High-Order Approximations for a Pseudoparabolic Equation of Turbulent Mass-Transfer Diffusion

Abstract

Numerical solutions of turbulent mass-transfer diffusion present challenges due to the nonlinearity of the elliptic–parabolic degeneracy of the mathematical models. Our main result in this paper concerns the development and implementation of an efficient high-order numerical method that is fourth-order accurate in space and second-order accurate in time for computing both the solution and its gradient for a Barenblatt-type equation. First, we reduce the original Neumann boundary value problem to a Dirichlet problem for the equation of the solution gradient. This problem is then solved by a compact fourth-order spatial approximation. To implement the numerical discretization, we employ Newton’s iterative method. Then, we compute the original solution while preserving the order of convergence. Numerical test results confirm the efficiency and accuracy of the proposed numerical scheme.

1. Introduction

The foundational principles of seepage (filtration) theory for homogeneous fluids in fractured rock formations were introduced by Barenblatt, Zhaltov, and Kochina [1]. Unlike the classical filtration theory for porous media, the innovative aspect of the approach presented in [1,2,3,4] lies in the introduction of two distinct fluid pressures at each spatial point, one of which is associated with the pores and the other with the fissures. This framework enables the investigation of fluid exchange between the fissures and the pore network. In the simplified scenario where fluid pressure is considered, this leads to the formulation of the so-called fissured equation, which belongs to the class of linear pseudoparabolic equations:

$${\displaystyle \frac{\partial u}{\partial t}}+{\mathcal{L}}_{1}u+{\displaystyle \frac{\partial }{\partial t}}{\mathcal{L}}_{2}u=0$$

(1)

where ${\mathcal{L}}_1$ and ${\mathcal{L}}_2$ are elliptic operators. Equations of this form (1) feature different linear or nonlinear differential operators ${\mathcal{L}}_1$ and ${\mathcal{L}}_2$, and serve as mathematical representations of phenomena such as heat conduction [5], wave propagation [6], quasi-static processes in semiconductors and magnetics [7], and filtration of two-phase flows in porous media with dynamic capillary pressure [8].

Linear pseudoparabolic equations have been expensively investigated in the literature both analytically and numerically; see, e.g., [9,10,11,12]. Nonlinear pseudoparabolic equations have been studied by many authors. Results for the existence and uniqueness of solution were obtained in [13,14,15]. We refer the reader to the review papers in [7,16] for analytical results and applications of nonlinear pseudoparabolic equations.

The Benjamin–Bona–Mahony (BBM) and Benjamin–Bona–Mahony–Burgers (BBMB) equations are both important classes of nonlinear pseudoparabolic equations.

BBM equations or generalized BBM equations have been studied in many papers, i.e., Equation (1), where

$${\mathcal{L}}_{1}u=(1+u^p){\displaystyle \frac{\partial u}{\partial x}},{\mathcal{L}}_{2}u=-{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}},p>0.$$

For example, the properties of the solution to this equation were studied in [17], exact solutions were obtained in [18], and high-order finite difference approximations were constructed in [19,20].

BBMB [6] equations are represented by (1), where

$${\displaystyle {\mathcal{L}}_{1}u=(1+u)\frac{\partial u}{\partial x}-\frac{{\partial }^{2}u}{\partial x^2},{\mathcal{L}}_{2}u=-\frac{{\partial }^{2}u}{\partial x^2}.}$$

These equations play a key role in the study of long waves in nonlinear dispersive systems and other fields of applied mathematics; see [21]. Results for the existence and uniqueness of the solution were presented in [22]. A wide range of numerical techniques have been introduced in the literature to solve BBMB-type problems. For example, second-order numerical methods were developed in [23,24,25], while fourth-order spatial discretizations were developed in [26,27]. The authors of [27] applied the improved cubic B-spline collocation technique to solve a 1D BBMB equation, while in [26] the authors developed a three-level linear-implicit difference scheme for 2D BBMB equations. Furthermore, a Strang splitting and quintic B-spline collocation scheme was developed in [28].

The generalized BBMB equation (GBBMB) (1), where

$${\mathcal{L}}_{1}u={\displaystyle \frac{\partial u}{\partial x}}-{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}-{\displaystyle \frac{\partial G\left(u\right)}{\partial x}},{\mathcal{L}}_{2}u=-{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}},$$

for a given function $G(·)$, was studied in [29,30,31,32]. In [29,30], the authors proposed numerical approaches for solving 1D and 2D GBBMB equations, achieving first-order temporal accuracy and second-order spatial accuracy. They utilized forward finite differences for time discretization, while spatial approximation was handled through Kansa’s method [29] and the interpolating element-free Galerkin method [30]. The time discretization in [31,32] relied on finite difference schemes, whereas spatial derivatives were approximated in [31] using the Legendre spectral element methods and in [32] using a combination of Lucas and Fibonacci polynomials. Polynomial functions with an embedded parameter were used in [33] to approximate the 1D and 2D GBBMB equations. In [34], the authors applied a spectral meshless radial point interpolation to solve the 2D BBBMB equation.

The authors of [35] developed and analyzed a numerical scheme for the Dirichlet initial boundary value problem associated with nonlinear pseudoparabolic equations. Their approach employed a spectral collocation method for spatial discretization, with Jacobi polynomials utilized for approximation. The Neumann problem for nonlinear pseudoparabolic equations of type (1)

$$L_{1}u=-{\displaystyle \frac{{\partial }^{2}G\left(u\right)}{\partial x^2}},{\mathcal{L}}_{2}u=-{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}$$

describing the aggregate population recovery was studied in [36]. The authors investigated the existence and uniqueness of a global solution along with the regularity properties and instability conditions of steady-state solutions for the 2D version of the problem.

Another class of nonlinear pseudobarabolic equations considered in the literature is (1) for

$$L_{1}u=-{\displaystyle \frac{{\partial }^2{G}_1\left(u\right)}{\partial x^2}},{\mathcal{L}}_{2}u=-{\displaystyle \frac{{\partial }^2{G}_2\left(u\right)}{\partial x^2}},$$

(2)

where $G_1(·)$ and $G_2(·)$ are given functions. In [37], the existence and uniqueness of a smooth solution for a pseudoparabolic equation related to nonlinear one-dimensional viscoelasticity was established. Short-time existence of solutions with constant compact support of the nonlinear pseudoparabolic equation was proved in [38]. In [39], the authors investigated a pseudoparabolic regularization of a viscous diffusion equation, focusing on the global existence of solutions and their stabilization to a steady state.

In this paper, we consider the Barenblatt-type Equations (1) and (2). In (2), the function u is replaced by the gradient ${\displaystyle \frac{\partial u}{\partial x}}$. To the best of our knowledge, this equation has not previously been studied numerically. In [2], the authors used an implicit scheme of order $O\left(\Delta t+h^2\right)$, where $\Delta t$ and h are time and space mesh step sizes, to validate their theoretical results. Here, we construct and investigate a new high-order numerical scheme that is second-order accurate in time and fourth-order accurqate in space for computing the solution u and its gradient.

The rest of this paper is organized as follows: Section 2 introduces the famous Barenblatt model equation and discusses its various physical properties; in Section 3, the well-posedness of the model problem and a reduced formulation are discussed; Section 4 focuses on deriving a discrete scheme with accuracy $O\left({(\Delta t)}^2+h^4\right)$; Section 5 presents Newton’s iterative method for solving the difference schemes; and computational test examples are provided in Section 6, after which the paper concludes with a summary and conclusions.

2. The Model Differential Problem

In [2], the authors considered an initial boundary value problem (IBVP) for the equation

$${\displaystyle \frac{\partial u}{\partial t}}={\displaystyle \frac{\partial }{\partial x}}\left(\phi \left({\displaystyle \frac{\partial u}{\partial x}}\right)\right)+\tau {\displaystyle \frac{{\partial }^2}{\partial t\partial x}}\left(\psi \left({\displaystyle \frac{\partial u}{\partial x}}\right)\right)+f(x,t),$$

(3)

where $\tau$ represents a small positive parameter and $\phi ,\psi :\mathbb{R}\to \mathbb{R}$ are given functions. The function $\phi$ is non-monotonic, whereas $\psi$ is strictly increasing and remains uniformly bounded over $\mathbb{R}$. Additionally, as $p\to \pm \infty$, the derivative of $\phi$ satisfies the asymptotic relation $|{\phi }^{\prime }\left(p\right)|\to 0$.

Equation (3) serves as a mathematical model for turbulent heat or mass transfer in stably stratified shear flows, where $\partial u/\partial x$ is non-negative. In such cases, $\phi \left(p\right)>0$ for $p>0$, and the function satisfies the condition $\phi \left(0\right)=\phi (+\infty )=0$.

In [40], the well-posedness of the initial boundary value problem (IBVP) associated with (3) was established; moreover, the qualitative properties of the solutions were analyzed in a specific model case. The observed behavior of the solutions aligned with both experimental data and numerical simulations.

In general, one-dimensional processes are described by the conservation law

$${\displaystyle \frac{\partial u}{\partial t}}+{\displaystyle \frac{\partial q}{\partial x}}=0,q=-k{\displaystyle \frac{\partial u}{\partial x}},$$

(4)

where q represents the heat or mass flux, x is the space coordinate within a given real interval, $t>0$ denotes time, and k is the diffusivity. If k is a given function $k_0$ of $u(x,t)$, ${\displaystyle \frac{\partial u}{\partial x}(x,t)}$, x, and t, then its non-negativity does not necessarily ensure the parabolicity of (4), as the product ${\displaystyle \frac{\partial u}{\partial x}{k}_0\left(u,\frac{\partial u}{\partial x},x,t\right)}$ can change its sign; see, e.g., [1,2,3,4,40].

In [3], the authors studied a mathematical model for heat or mass transfer in stably stratified turbulent shear sheer flows, where the temperature (respectively, the concentration u) satisfies an equation of type (4).

Under constant external conditions, the steady-state diffusivity, i.e., the diffusivity in a state of mechanical and thermal equilibrium, depends solely on the gradient

$$k(x,t)=k_0\left({\displaystyle \frac{\partial u}{\partial x}}(x,t)\right).$$

(5)

In the case of a stably stratified turbulent flow, the effective temperature or mass diffusivity $k_0$ diminishes rapidly when the temperature gradient (or equivalently, the concentration gradient) reaches high values. The common representation of $k_0$ is provided by

$$k_0\left(p\right)={\displaystyle \frac{A}{B+p^2}},A,B>0,$$

(6)

which ensures that the function $\phi \left(p\right)=p{k}_0\left(p\right)$ exhibits a monotonic behavior, that is, it increases for $0<p<\alpha$ for some critical threshold $\alpha >0$ and decreases for $p>\alpha$.

More generally, the authors of [4] considered smooth functions $\phi$ satisfying the following conditions for some $\alpha >0$:

$$\phi \left(0\right)=\phi (+\infty )=0,0<\phi \left(p\right)\le \phi \left(\alpha \right),\mathrm{for}p>0.$$

(7)

After substituting (5) into the balance law (4), the resulting second-order partial differential equation

$${\displaystyle \frac{\partial u}{\partial t}}={\displaystyle \frac{\partial }{\partial x}}\left(\phi \left({\displaystyle \frac{\partial u}{\partial x}}\right)\right)$$

(8)

is not of forward parabolic type at points where ${\phi }^{\prime }\left({\displaystyle \frac{\partial u}{\partial x}}\right)<0$. This may lead to an ill-posed initial boundary value problem (IBVP). It was shown in [41] that if $\psi (·)\equiv 1$ in Equation (3), then its solution converges to the solution of a parabolic equation of form (8) as $\tau \to 0$.

3. Well-Posedness of the Original Model and the Reduced Problem

Our main result concerns the construction and implementation of an efficient high-order numerical method for solving Equation (3) in the domain $(x,t)\in Q_T=\Omega \times (0,T]$, $\Omega =(0,1)$ subject to the initial condition

$$u(x,0)=u^0\left(x\right),0<x<1$$

(9)

and the boundary conditions

$${\displaystyle \frac{\partial u}{\partial x}}(0,t)=u_l\left(t\right),{\displaystyle \frac{\partial u}{\partial x}}(1,t)=u_r\left(t\right),t\in (0,T].$$

(10)

The existence and uniqueness of solutions in appropriate functional spaces for the IBVP of Equation (8), including the behavior of the function $\phi (·)$ of type (5)–(7), have been studied in [40]; see, e.g., Section 1.4.

Here, we use the following functional spaces. Let $L_p(\Omega )(1\le p\le \infty )$ be the Banach space of the measurable in $\Omega$ functions such that

$${\parallel u\parallel }_{L_p(\Omega )}={\left({\int }_{\Omega }{\left|u\left(k\right)\right|}^{p}dx\right)}^{1/p}{<\infty \mathrm{and}\parallel u\parallel }_{L_{\infty }(\Omega )}=\mathrm{vrai}\underset{x\in \Omega }{sup}\left|u\left(x\right)\right|<\infty .$$

Next, let $W_p^l(\Omega )$ be the Banach space of functions from $L_p(\Omega )$ with generalized derivatives existing up to order l, with the norm defined as follows:

$${\parallel u\parallel }_{W_p^l(\Omega )}={\left[{\int }_{\Omega }{\left(\right|u|}^p+\sum _{k=1}^l\sum _{\left(k\right)}\parallel D^{\left(k\right)}u{\parallel }^{p}dx)\right]}^{1/p}<\infty$$

where $D^{\left(k\right)}u$ is a derivative of order k. Similarly, let $C^l(\Omega )(l=0,1,\dots )$ be the space of functions that are continuous in $\Omega$ and have continuous derivatives up to order l in $\Omega$:

$$C^0(\Omega )={C(\Omega ),\parallel u\parallel }_{\Omega }^{\left(0\right)}={\parallel u\parallel }_{C(\Omega )}=\underset{x\in \Omega }{sup}\left|u\left(x\right)\right|,$$

$${\left|u\right|}_{\Omega }^l={\parallel u\parallel }_{C^l(\Omega )}={\parallel u\parallel }_{C(\Omega )}+\sum _{\left|k\right|\le l}{\parallel D^{\left(k\right)}u\parallel }_{C(\Omega )}.$$

Finally, let $W_p^l(0,T;X)$ (X be a Banach space and let $l=0,1,\dots$) be the Banach space of the measurable functions from $[0,T]$ into X, which have generalized derivatives

$${\parallel u\parallel }_{W_p^l(0,T;X)}=\sum _{k=0}^l{∥{\displaystyle \frac{{\partial }^{k}u}{\partial t^k}}∥}_{L_p(0,T;X)}.$$

Now, from [40], we apply Theorem 1.1 (page 118) and Theorem 1.3 (page 137) to the problem in (3), (9), (10).

We assume the following conditions concerning the functions $\phi$, $\psi$:

A1.

$\psi \left(S\right)\subset C^2\left(\mathbb{R}\right)$, ${\displaystyle {\psi }^{\prime }\left(S\right)=\frac{d\psi }{dS}\ge \delta >0.}$

A2.

$\phi \left(S\right)\subset C^1\left(\mathbb{R}\right)$, $-{\phi }^{\prime }\left(S\right)\le C(1+{\psi }^{\prime }\left(S\right)),C>0.$

Direct application of the mentioned theorems from [40] yields the result.

Theorem 1.

Let $u^0\in W_2^2(\Omega )$, $u_l,u_r\in C^1(0,T)$ and the functions φ, ψ satisfy conditions A1 and A2. Then, for any function $f\in L_2\left(Q_T\right)$, there exists a unique solution $u(x,t)$ such that $u\in L_{\infty }(0,T;W_2^2(\Omega ))$, ${\displaystyle \frac{\partial u}{\partial t}\in L_{\infty }(0,T;W_2^2(\Omega ))}$, and ${\displaystyle \frac{{\partial }^{3}u}{\partial x^2\partial t}\in L_2\left(Q_T\right)}$ for the problem in (3), (9), (10).

This solution is referred to as a regular solution in [40], as it satisfies (3) almost everywhere in $Q_T$ and fulfills the initial and boundary conditions in the classical sense.

Let

$$v={\displaystyle \frac{\partial u}{\partial x}};$$

(11)

then, differentiating Equation (3) with respect to x and using (11), we obtain

$${\displaystyle \frac{\partial v}{\partial t}}={\displaystyle \frac{{\partial }^2\phi \left(v\right)}{\partial x^2}}+\tau {\displaystyle \frac{{\partial }^3\psi \left(v\right)}{\partial t\partial x^2}}+{\displaystyle \frac{\partial f(x,t)}{\partial x}}.$$

(12)

Corresponding to (9), (10), the initial and boundary conditions after substituting (11) become

$$\begin{array}{c}\hfill \begin{array}{cc}& v(x,0)=v^0\left(x\right),v^0\left(x\right)={\displaystyle \frac{d{u}^0\left(x\right)}{dx}},0<x<1,\hfill \\ & v(0,t)=u_l\left(t\right),v(1,t)=u_r\left(t\right),t\in (0,T].\hfill \end{array}\end{array}$$

(13)

Then, the existence and uniqueness of the solution $v(x,t)$ of the reduced problem in (12), (13) in the corresponding functional spaces follows from Theorem 1. Indeed, the smoothness conditions on the input data proposed in Theorem 1 ensure the existence of a solution $v\in L_{\infty }(0,T;W_2^1(\Omega ))$ for the in problem (12), (13). However, in the discretization process presented in the next section, we assume the existence and uniqueness of a solution that possesses continuous derivatives up to fourth order in space and second order in time.

4. O(($\mathbf{▵}\mathit{t}$)² + h⁴) Discretization

In this section, we develop a numerical method of fourth-order accuracy in space and second-order accuracy in time for solving the problem in (3), (9), (10).

We consider the uniform spatial mesh with a number of nodes I:

$${\overline{w}}_h=\{x_i=ih,i=0,1,\dots ,I,h=1/I\}$$

and introduce the following notation: $v_i=v(x_i,t)$, ${\phi }_i=\phi \left(v_i\right)$, ${\psi }_i=\psi \left(v_i\right)$.

4.1. Discretization of the Reduced Problem

The concept for developing this approach stems from the following well-known approximation:

$${\nu }_{\overline{x}x,i}={\displaystyle \frac{\nu \left(x_{i-1}\right)-2v\left(x_i\right)+v\left(x_{i+1}\right)}{h^2}}={\displaystyle \frac{d^{2}v}{d{x}^2}}\left(x_i\right)+{\displaystyle \frac{h^2}{12}}{\displaystyle \frac{d^{4}v}{d{x}^4}}\left(x_i\right)+O\left(h^4\right),x_i\in {\overline{w}}_h.$$

(14)

In view of (14), Equation (12) is discretized as follows:

$${\displaystyle \frac{\partial v_i}{\partial t}}={\left(\phi \left(v\right)\right)}_{\overline{x}x,i}-{\displaystyle \frac{h^2}{12}}{\displaystyle \frac{{\partial }^4\phi \left(v\right)}{\partial x^4}}{|}_{x=x_i}+\tau {\displaystyle \frac{\partial }{\partial t}}\left({\left(\psi \left(v\right)\right)}_{\overline{x}x,i}-{\displaystyle \frac{h^2}{12}}{\displaystyle \frac{{\partial }^4\psi \left(v\right)}{\partial x^4}}{|}_{x=x_i}\right)+{\displaystyle \frac{\partial f(x,t)}{\partial x}}{|}_{x=x_i}.$$

(15)

Next, we differentiate (12) twice with respect to x and obtain

$${\displaystyle \frac{\partial v^3}{\partial t\partial x^2}}={\displaystyle \frac{{\partial }^4\phi \left(v\right)}{\partial x^4}}+\tau {\displaystyle \frac{{\partial }^5\psi \left(v\right)}{\partial t\partial x^4}}+{\displaystyle \frac{{\partial }^{3}f(x,t)}{\partial x^3}}.$$

(16)

From (15) and (16), we obtain

$${\displaystyle \frac{\partial v_i}{\partial t}}={\left(\phi \left(v\right)\right)}_{\overline{x}x,i}+{\displaystyle \frac{\partial }{\partial t}}\left(\tau {\left(\psi \left(v\right)\right)}_{\overline{x}x,i}-{\displaystyle \frac{h^2}{12}}{\displaystyle \frac{\partial v^2}{\partial x^2}}{|}_{x=x_i}\right)+{\displaystyle \frac{h^2}{12}}{\displaystyle \frac{{\partial }^{3}f(x,t)}{\partial x^3}}{|}_{x=x_i}+{\displaystyle \frac{\partial f(x,t)}{\partial x}}{|}_{x=x_i}.$$

(17)

Further, we define a uniform temporal mesh with a number N of grid nodes:

$${\overline{w}}_{\Delta t}=\{t_n=n\Delta t,n=0,1,\dots ,N,n=T/N\}$$

and approximate the problem in (13), (17) by a second-order difference scheme in time:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{F}\left(v_i^{n+1}\right):=& {\displaystyle \frac{v_i^{n+1}-v_i^n}{\Delta t}}-{\displaystyle \frac{1}{2}}{\left({\phi }^{n+1}+{\phi }^n\right)}_{\overline{x}x,i}-{\displaystyle \frac{\tau }{\Delta t}}{\left({\psi }^{n+1}-{\psi }^n\right)}_{\overline{x}x,i}\hfill \\ & +{\displaystyle \frac{h^2}{12\Delta t}}{\left(v^{n+1}-v^n\right)}_{\overline{x}x,i}\hfill \\ & -{\displaystyle \frac{h^2}{24}}\left({\overline{\overline{f}}}_i^{n+1}+{\overline{\overline{f}}}_i^n\right)-{\displaystyle \frac{1}{2}}\left({\overline{f}}_i^{n+1}+{\overline{f}}_i^n\right)=0,i=1,2,\dots ,I-1,\hfill \\ & v_i^{n+1}=v^0\left(x_i\right),i=0,1,\dots ,I,\hfill \\ & v_0^{n+1}=u_l^{n+1},v_I^{n+1}=u_r^{n+1},\hfill \end{array}\end{array}$$

(18)

where $v_i^n=v(x_i,t_n)$, ${\phi }_i^n=\phi \left(v_i^n\right)$, ${\psi }_i^n=\psi \left(v_i^n\right)$, ${\displaystyle {\overline{f}}_i^n=\frac{\partial f(x,t)}{\partial x}{|}_{x=x_i,t=t_n}}$ and ${\displaystyle {\overline{\overline{f}}}_i^n=\frac{{\partial }^{3}f(x,t)}{\partial x^3}{|}_{x=x_i,t=t_n}}$.

As a result of the above construction, the following assertion holds.

Theorem 2.

Based on Theorem 1, we assume that the following conditions are satisfied:

(i)

$v^0\left(x\right)\in C^3(\Omega )$, $u_l,u_r\in C^3(0,T)$, and ${\displaystyle \frac{{\partial }^{3}f(x,t)}{\partial x^3}\in L^2(\Omega )}$.

(ii)

The functions φ and ψ possess continuous derivatives up to the fourth order in space and second order in time.

Then, the solution of the problem in (12), (13) is assumed to possesses continuous derivatives up to the fourth order in space and second order in time, which is necessary for the construction of the approximation in (18). This difference scheme approximates the solution of the differential problem in (12), (13) with second-order accuracy in time and fourth-order accuracy in space, i.e., the truncation error is $O\left({(\Delta t)}^2+h^4\right)$.

Proof. 

The proof follows from the discretization procedure in (14)–(18). □

In this way, we obtain the numerical solution v with accuracy $O\left({(\Delta t)}^2+h^4\right)$. It remains to determine the solution u while maintaining this accuracy.

4.2. Discretization of the Original Problem

Because v is known, we can linearize Equation (3) as follows:

$${\displaystyle \frac{\partial u}{\partial t}}={\phi }^{\prime }\left(v\right){\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+\tau {\displaystyle \frac{\partial }{\partial t}}\left({\psi }^{\prime }\left(v\right){\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}\right)+f(x,t).$$

(19)

Then, to solve the linear pseudoparabolic problem in (9), (10), (19), we can apply a simple approach involving directly replacing the spatial derivatives with fourth-order approximations and Crank–Nicolson timestepping. In this way, we obtain the following discretization:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \frac{u_i^{n+1}-u_i^n}{\Delta t}}=& {\displaystyle \frac{1}{2}}{\phi }^{\prime }\left(v_i^{n+1}\right)u_{\mathring{x}\mathring{x},i}^{n+1}+{\displaystyle \frac{1}{2}}{\phi }^{\prime }\left(v_i^n\right)u_{\mathring{x}\mathring{x},i}^n+\tau \left({\psi }^{\prime }\left(v_i^{n+1}\right)u_{\mathring{x}\mathring{x},i}^{n+1}-{\psi }^{\prime }\left(v_i^n\right)u_{\mathring{x}\mathring{x},i}^n\right)\hfill \\ & +{\displaystyle \frac{1}{2}}\left(f_i^{n+1}+f_i^n\right),i=2,3,\dots ,I-2,\hfill \\ \hfill {\displaystyle \frac{u_1^{n+1}-u_1^n}{\Delta t}}=& {\displaystyle \frac{1}{2}}{\phi }^{\prime }\left(v_1^{n+1}\right)u_{\widehat{x}\widehat{x},1}^{n+1,+}+{\displaystyle \frac{1}{2}}{\phi }^{\prime }\left(v_1^n\right)u_{\widehat{x}\widehat{x},1}^{n,+}\hfill \\ & +\tau \left({\psi }^{\prime }\left(v_1^{n+1}\right)u_{\widehat{x}\widehat{x},1}^{n+1,+}-{\psi }^{\prime }\left(v_1^n\right)u_{\widehat{x}\widehat{x},1}^{n,+}\right)+{\displaystyle \frac{1}{2}}\left(f_1^{n+1}+f_1^n\right),\hfill \\ \hfill {\displaystyle \frac{u_{I-1}^{n+1}-u_{I-1}^n}{\Delta t}}=& {\displaystyle \frac{1}{2}}{\phi }^{\prime }\left(v_{I-1}^{n+1}\right)u_{\widehat{x}\widehat{x},I-1}^{n+1,-}+{\displaystyle \frac{1}{2}}{\phi }^{\prime }\left(v_{I-1}^n\right)u_{\widehat{x}\widehat{x},I-1}^{n,-}\hfill \\ & +\tau \left({\psi }^{\prime }\left(v_{I-1}^{n+1}\right)u_{\widehat{x}\widehat{x},I-1}^{n+1,-}-{\psi }^{\prime }\left(v_{I-1}^n\right)u_{\widehat{x}\widehat{x},I-1}^{n,-}\right)+{\displaystyle \frac{1}{2}}\left(f_{I-1}^{n+1}+f_{I-1}^n\right),\hfill \\ & u_{\widehat{x},0}^{n+1,+}=v_0^{n+1},u_{\widehat{x},I}^{n+1,-}=v_I^{n+1}\hfill \end{array}\end{array}$$

(20)

where

$$\begin{array}{ccc}& & u_{\mathring{x}\mathring{x},i}^n=-{\displaystyle \frac{u_{i-2}^n+u_{i+2}^n}{12h^2}}+{\displaystyle \frac{4(u_{i-1}^n+u_{i+1}^n)}{3h^2}}-{\displaystyle \frac{5}{2h^2}}{u}_i^n,\hfill \\ & & u_{\widehat{x}\widehat{x},i}^{n,\pm }={\displaystyle \frac{15u_i^n}{4h^2}}-{\displaystyle \frac{77u_{i\pm 1}^n}{6h^2}}+{\displaystyle \frac{107u_{i\pm 2}^n}{16h^2}}-{\displaystyle \frac{13u_{i\pm 3}^n}{h^2}}+{\displaystyle \frac{61u_{i\pm 4}^n}{12h^2}}-{\displaystyle \frac{5u_{i\pm 5}^n}{6h^2}},\hfill \\ & & u_{\widehat{x},i}^{n,\pm }=\mp {\displaystyle \frac{25u_i^n}{12h}}\pm {\displaystyle \frac{4u_{i\pm 1}^n}{h}}\mp {\displaystyle \frac{3u_{i\pm 2}^n}{h}}\pm {\displaystyle \frac{4u_{i\pm 3}^n}{3h}}\mp {\displaystyle \frac{u_{i\pm 4}^n}{4h}}.\hfill \end{array}$$

5. Implementation of the Difference Scheme

In this section, we discuss the solution of the nonlinear system in (18). To this end, we apply Newton’s iterative process and establish its quadratic convergence.

Let $u^{\left(k\right)}$ denote the approximation to $u^{n+1}$ at the k-th iteration. At each time layer, we solve the linearized system for $k=0,1,\dots$:

$$\begin{array}{c}\hfill \begin{array}{cc}& {\displaystyle \frac{v_i^{(k+1)}-v_i^n}{\Delta t}}-{\displaystyle \frac{1}{2}}{\left({\left({\phi }^{\prime }\right)}^{\left(k\right)}{v}^{(k+1)}\right)}_{\overline{x}x,i}-{\displaystyle \frac{\tau }{\Delta t}}{\left({\left({\psi }^{\prime }\right)}^{\left(k\right)}{v}^{(k+1)}\right)}_{\overline{x}x,i}+{\displaystyle \frac{h^2}{12\Delta t}}{v}_{\overline{x}x,i}^{(k+1)}\hfill \\ & =-{\displaystyle \frac{1}{2}}{\left({\left({\phi }^{\prime }\right)}^{\left(k\right)}{v}^{\left(k\right)}\right)}_{\overline{x}x,i}-{\displaystyle \frac{\tau }{\Delta t}}{\left({\left({\psi }^{\prime }\right)}^{\left(k\right)}{v}^{\left(k\right)}\right)}_{\overline{x}x,i}+{\displaystyle \frac{1}{2}}{\left({\phi }^{\left(k\right)}+{\phi }^n\right)}_{\overline{x}x,i}\hfill \\ & +{\displaystyle \frac{\tau }{\Delta t}}{\left({\psi }^{\left(k\right)}-{\psi }^n\right)}_{\overline{x}x,i}+{\displaystyle \frac{h^2}{12\Delta t}}{v}_{\overline{x}x,i}^n+{\displaystyle \frac{h^2}{24}}\left({\overline{\overline{f}}}_i^{n+1}+{\overline{\overline{f}}}_i^n\right)\hfill \\ & +{\displaystyle \frac{1}{2}}\left({\overline{f}}_i^{n+1}+{\overline{f}}_i^n\right),i=1,2,\dots ,I-1,\hfill \\ & v_0^{(k+1)}=u_l^{n+1},v_I^{(k+1)}=u_r^{n+1}.\hfill \\ & v_i^{\left(0\right)}=v_i^n,i=0,1,\dots ,I\hfill \end{array}\end{array}$$

(21)

where we use the notation ${\displaystyle {\phi }^{\prime }=\frac{\partial \phi }{\partial v}}$, ${\displaystyle {\psi }^{\prime }=\frac{\partial \psi }{\partial v}}$.

The iteration process continues until to reaching the desires accuracy $ϵ$, i.e., $\parallel v^{(k+1)}-v^{\left(k\right)}\parallel \le ϵ$. Then, we set $v^{n+1}:=v^{(k+1)}$.

We define the discrete norm $\parallel v\parallel =\underset{0\le i\le I}{max}\left|v_i\right|$.

Theorem 3.

Let the conditions of Theorems 1 and 2 be fulfilled and let ${\displaystyle \frac{\Delta t}{h^2}}\parallel {\phi }^{″}\parallel +{\displaystyle \frac{2\tau }{h^2}}\parallel {\psi }^{″}\parallel \le M$, where M is a positive constant. Then, at each time layer, $v^{(k+1)}$ converges to $v^{n+1}$ quadratically for sufficiently large k, i.e.,

$$\parallel v^{(k+1)}-v^{n+1}\parallel \le M\parallel v^{\left(k\right)}-v^{n+1}{\parallel }^2.$$

Proof. 

The Newton iteration process (21) can be written as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}& {\displaystyle \frac{{\delta }_i^{(k+1)}}{\Delta t}}-{\displaystyle \frac{1}{2}}{\left({\left({\phi }^{\prime }\right)}^{\left(k\right)}{\delta }^{(k+1)}\right)}_{\overline{x}x,i}-{\displaystyle \frac{\tau }{\Delta t}}{\left({\left({\psi }^{\prime }\right)}^{\left(k\right)}{\delta }^{(k+1)}\right)}_{\overline{x}x,i}+{\displaystyle \frac{h^2}{12\Delta t}}{v}_{\overline{x}x,i}^{(k+1)}=-\mathcal{F}\left(v_i^{\left(k\right)}\right),\hfill \\ & {\delta }_0^{(k+1)}=u_l^{n+1}-v_0^{\left(k\right)},{\delta }_I^{(k+1)}=u_r^{n+1}-v_I^{\left(k\right)}\hfill \end{array}\end{array}$$

where ${\delta }_i^{(k+1)}=v_i^{(k+1)}-v_i^{\left(k\right)}$, $i=0,1,\dots ,I$. The corresponding matrix-vector form is

$$\mathcal{M}{\delta }^{(k+1)}=-\tilde{\mathcal{F}}\left(v^{\left(k\right)}\right),$$

(22)

where $v^{\left(k\right)}={\left[v_1^{\left(k\right)},v_2^{\left(k\right)},\dots ,v_I^{\left(k\right)}\right]}^T$, ${\delta }^{\left(k\right)}={\left[{\delta }_1^{\left(k\right)},{\delta }_2^{\left(k\right)},\dots ,{\delta }_I^{\left(k\right)}\right]}^T$, $\tilde{\mathcal{F}}\left(v^{\left(k\right)}\right)={\left[v_0^{\left(k\right)}-u_l^{n+1},\mathcal{F}\left(v_1^{\left(k\right)}\right),\dots ,\mathcal{F}\left(v_{I-1}^{\left(k\right)}\right),v_I^{\left(k\right)}-u_r^{n+1}\right]}^T$ and $\mathcal{M}={\left(m_{i,j}\right)}_{0\le i,j\le I}$

$$m_{i,j}=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\Delta t}+\frac{1}{h^2}{\left({\phi }_i^{\prime }\right)}^{\left(k\right)}+\frac{2\tau }{h^2\Delta t}{\left({\psi }_i^{\prime }\right)}^{\left(k\right)}-\frac{1}{6\Delta t},}\hfill & i=j,i\ne \{0,I\},\hfill \\ {\displaystyle -\frac{1}{2h^2}{\left({\phi }_{i+1}^{\prime }\right)}^{\left(k\right)}-\frac{\tau }{h^2\Delta t}{\left({\psi }_{i+1}^{\prime }\right)}^{\left(k\right)}+\frac{1}{12\Delta t},}\hfill & j=i+1,j>0,\hfill \\ {\displaystyle -\frac{1}{2h^2}{\left({\phi }_{i-1}^{\prime }\right)}^{\left(k\right)}-\frac{\tau }{h^2\Delta t}{\left({\psi }_{i-1}^{\prime }\right)}^{\left(k\right)}+\frac{1}{12\Delta t},}\hfill & j=i-1,j<I,\hfill \\ 1,\hfill & i=j=0,i=j=I,\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Let $e^{\left(k\right)}=v^{\left(k\right)}-v^{n+1}$. Applying Taylor series expansion in (18), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{F}\left(v_i^k-e_i^k\right)=& {\displaystyle \frac{v_i^{\left(k\right)}-e^{\left(k\right)}-v_i^n}{\Delta t}}-{\displaystyle \frac{1}{2}}{\left({\phi }^{\left(k\right)}-{\left({\phi }^{\prime }\right)}^{\left(k\right)}{e}^{\left(k\right)}+{\displaystyle \frac{{\left({\phi }^{″}\right)}^{\left(k\right)}}{2}}{\left(e^{\left(k\right)}\right)}^2+{\phi }^n\right)}_{\overline{x}x,i}\hfill \\ & -{\displaystyle \frac{\tau }{\Delta t}}{\left({\psi }^{\left(k\right)}-{\left({\psi }_i^{\prime }\right)}^{\left(k\right)}{e}^{\left(k\right)}+{\displaystyle \frac{{\left({\psi }^{″}\right)}^{\left(k\right)}}{2}}{\left(e^{\left(k\right)}\right)}^2-{\psi }^n\right)}_{\overline{x}x,i}\hfill \\ & +{\displaystyle \frac{h^2}{12\Delta t}}{\left(v^{\left(k\right)}-e^{\left(k\right)}-v^n\right)}_{\overline{x}x,i}\hfill \\ & -{\displaystyle \frac{h^2}{24}}\left({\overline{\overline{f}}}_i^{n+1}+{\overline{\overline{f}}}_i^n\right)-{\displaystyle \frac{1}{2}}\left({\overline{f}}_i^{n+1}+{\overline{f}}_i^n\right)=0,i=1,2,\dots ,I-1.\hfill \end{array}\end{array}$$

(23)

Then, we can rewrite (23) in the following form:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\displaystyle \frac{v_i^{\left(k\right)}-v_i^n}{\Delta t}}-{\displaystyle \frac{1}{2}}{\left({\phi }^{\left(k\right)}+{\phi }^n\right)}_{\overline{x}x,i}-{\displaystyle \frac{\tau }{\Delta t}}{\left({\psi }^{\left(k\right)}-{\psi }^n\right)}_{\overline{x}x,i}+{\displaystyle \frac{h^2}{12\Delta t}}{\left(v^{\left(k\right)}-v^n\right)}_{\overline{x}x,i}\hfill \\ & -{\displaystyle \frac{h^2}{24}}\left({\overline{\overline{f}}}_i^{n+1}+{\overline{\overline{f}}}_i^n\right)-{\displaystyle \frac{1}{2}}\left({\overline{f}}_i^{n+1}+{\overline{f}}_i^n\right)\hfill \\ & {\displaystyle \frac{-e_i^{\left(k\right)}}{\Delta t}}-{\displaystyle \frac{1}{2}}{\left(-{\left({\phi }^{\prime }\right)}^{\left(k\right)}{e}^{\left(k\right)}\right)}_{\overline{x}x,i}-{\displaystyle \frac{\tau }{\Delta t}}{\left(-{\left({\psi }^{\prime }\right)}^{\left(k\right)}{e}^{\left(k\right)}\right)}_{\overline{x}x,i}+{\displaystyle \frac{h^2}{12\Delta t}}{\left(-e^{\left(k\right)}\right)}_{\overline{x}x,i}\hfill \\ & -{\displaystyle \frac{1}{4}}{\left({\left({\phi }^{″}\right)}^{\left(k\right)}{\left(e^{\left(k\right)}\right)}^2\right)}_{\overline{x}x,i}-{\displaystyle \frac{\tau }{2\Delta t}}{\left({\left({\psi }^{″}\right)}^{\left(k\right)}{\left(e^{\left(k\right)}\right)}^2\right)}_{\overline{x}x,i}=0,i=1,2,\dots ,I-1.\hfill \end{array}\end{array}$$

(24)

Incorporating the boundary conditions, (24) is equivalent to

$$\begin{array}{c}\hfill \tilde{\mathcal{F}}\left(v^{\left(k\right)}\right)-\mathcal{M}\left(v^{\left(k\right)}-v^{n+1}\right)-\mathcal{G}\left({\left(e^k\right)}^2\right)=0,\end{array}$$

(25)

where ${\displaystyle \mathcal{G}\left({\left(e^k\right)}^2\right)={\left[0,{\mu }_1^{\left(k\right)},{\mu }_2^{\left(k\right)}\dots ,{\mu }_{I-1}^{\left(k\right)},0\right]}^T}$ and ${\displaystyle {\mu }_i^{\left(k\right)}=\frac{1}{4}{\left({\left({\phi }_i^{″}\right)}^{\left(k\right)}{\left(e_i^{\left(k\right)}\right)}^2\right)}_{\overline{x}x,i}+\frac{\tau }{2\Delta t}{\left({\left({\psi }_i^{″}\right)}^{\left(k\right)}{\left(e_i^{\left(k\right)}\right)}^2\right)}_{\overline{x}x,i}}$.

From (22) and (25), we derive

$$v^{\left(k\right)}-v^{n+1}=M^{-1}\left(\tilde{\mathcal{F}}\left(v^{\left(k\right)}\right)-\mathcal{G}\left({\left(e^k\right)}^2\right)\right)=-{\delta }^{(k+1)}-M^{-1}\mathcal{G}\left({\left(e^k\right)}^2\right).$$

Therefore, we obtain

$$v^{(k+1)}-v^{n+1}=-{\mathcal{M}}^{-1}\mathcal{G}\left({\left(e^k\right)}^2\right).$$

Taking into account the assumptions of the theorem and the diagonal dominance of the matrix $\mathcal{M}$, we can estimate

$$\parallel v^{(k+1)}-v^{n+1}\parallel \le \left({\displaystyle \frac{\Delta t}{h^2}}\parallel {\phi }^{″}\parallel +{\displaystyle \frac{2\tau }{h^2}}\parallel {\psi }^{″}\parallel \right)\parallel v^{\left(k\right)}-v^{n+1}{\parallel }^2.$$

Thus, the proof is completed. □

6. Numerical Tests

In this section, we illustrate the accuracy of the proposed numerical method in (18), (20) for solving the problem in (3), (9), (10). First, we estimate the accuracy and convergence rate in the maximal and $L_2$ discrete norms:

$$\begin{array}{cc}\hfill & {\displaystyle E_i\left(\nu \right)={\nu }_i^N-\nu (x_i,T),{\mathcal{E}}_{\infty }\left(\nu \right)={\mathcal{E}}_{\infty }^I\left(\nu \right)=\parallel E\left(\nu \right)\parallel ,C{R}_{\infty }\left(\nu \right)={log}_2\frac{{\mathcal{E}}_{\infty }^I\left(\nu \right)}{{\mathcal{E}}_{\infty }^{2I}\left(\nu \right)},}\\ & {\displaystyle {\mathcal{E}}_2\left(\nu \right)={\mathcal{E}}_2^I\left(\nu \right)=\sqrt{h\sum _{i=0}^I{E}_i^2\left(\nu \right)},C{R}_2\left(\nu \right)={log}_2\frac{{\mathcal{E}}_2^I\left(\nu \right)}{{\mathcal{E}}_2^{2I}\left(\nu \right)}.}\end{array}$$

The average number of iterations is denoted by $k_a$, and for the stopping criteria of Newton iteration process we take $ϵ$ = ${10}^{-10}$. To compute the problem in (20), we apply Jacobi preconditioning. The computations are performed for $T=0.5$.

To confirm the spatial and temporal accuracy of order $O\left({(\Delta t)}^2+h^4\right)$, we perform simulations using fixed ratios of $\Delta t=h^2$ and $\Delta t=h$, respectively.

Example 1.

Let

$$\begin{array}{cc}& {\displaystyle \tau =2,\phi \left(v\right)=v^2+1,\psi \left(v\right)=v^3,u_l\left(t\right)=2e^{-t/2},u_r\left(t\right)=2e^{-t/2+2},}\\ & {\displaystyle f(x,t)=e^{-t/2+2x}\left(-\frac{1}{2}-16e^{-t/2+2x}+144e^{-t+4x}\right),u^0\left(x\right)=e^{2x}.}\end{array}$$

Then, the exact solution of the problem in (3), (9), (10) is $u(x,t)=e^{-t/2+2x}$.

Table 1. Errors and order of convergence of v, $\Delta t=h^2$, Example 1.

I	${\mathcal{E}}_{\mathit{\infty }}\left(\mathit{v}\right)$	${\mathcal{CR}}_{\mathit{\infty }}\left(\mathit{v}\right)$	${\mathcal{E}}_2\left(\mathit{v}\right)$	${\mathcal{CR}}_2\left(\mathit{v}\right)$	${\mathit{k}}_{\mathit{a}}$
8	1.6000 $\times {10}^{-2}$		1.0309 $\times {10}^{-2}$		4.000
16	1.0265 $\times {10}^{-3}$	3.9624	6.6351 $\times {10}^{-4}$	3.9576	4.000
32	6.4642 $\times {10}^{-5}$	3.9891	4.1794 $\times {10}^{-5}$	3.9887	3.002
64	4.0481 $\times {10}^{-6}$	3.9972	2.6173 $\times {10}^{-6}$	3.9971	3.000
128	2.5311 $\times {10}^{-7}$	3.9994	1.6365 $\times {10}^{-7}$	3.9994	3.193

and

Table 2. Errors and order of convergence of u, $\Delta t=h^2$, Example 1.

I	${\mathcal{E}}_{\mathit{\infty }}\left(\mathit{u}\right)$	${\mathcal{CR}}_{\mathit{\infty }}\left(\mathit{u}\right)$	${\mathcal{E}}_2\left(\mathit{u}\right)$	${\mathcal{CR}}_2\left(\mathit{u}\right)$
8	2.3957 $\times {10}^{-3}$		1.7010 $\times {10}^{-3}$
16	9.2693 $\times {10}^{-4}$	1.3699	7.0057 $\times {10}^{-4}$	1.2798
32	7.6290 $\times {10}^{-5}$	3.6029	5.6056 $\times {10}^{-5}$	3.6436
64	5.2496 $\times {10}^{-6}$	3.8612	3.8064 $\times {10}^{-6}$	3.8804
128	3.3779 $\times {10}^{-7}$	3.9580	2.4282 $\times {10}^{-7}$	3.9705

respectively illustrate the spatial convergence of solutions v and u for a fixed ratio $\Delta t=h^2$. Our numerical experiments show that the order of convergence in space is four. Because the ratio between the time and space step sizes is fixed, we can deduce that the temporal convergence rate is at least two. In

Table 3. Errors and order of convergence of u, $\Delta t=h$, Example 1.

I	${\mathcal{E}}_{\mathit{\infty }}\left(\mathit{u}\right)$	${\mathcal{CR}}_{\mathit{\infty }}\left(\mathit{u}\right)$	${\mathcal{E}}_2\left(\mathit{u}\right)$	${\mathcal{CR}}_2\left(\mathit{u}\right)$	${\mathit{k}}_{\mathit{a}}$
8	9.8246 $\times {10}^{-2}$		7.2507 $\times {10}^{-2}$		6.000
16	2.4345 $\times {10}^{-2}$	2.0128	1.7866 $\times {10}^{-2}$	2.0209	5.000
32	5.9211 $\times {10}^{-3}$	2.0397	4.3382 $\times {10}^{-3}$	2.0421	5.000
64	1.4461 $\times {10}^{-3}$	2.0337	1.0590 $\times {10}^{-3}$	2.0343	4.000
128	3.5647 $\times {10}^{-4}$	2.0203	2.6101 $\times {10}^{-4}$	2.0206	4.206

, we present results for the solution u and $\Delta t=h$. The second-order accuracy observed in this table is due to the dominance of the lower-order convergence in this case with respect to the time variable. Consequently, the accuracy is $O\left({(\Delta t)}^2+h^4\right)$ for both of numerical solutions u and v. Moreover, convergence is achieved with a small number of iterations, and the average number of iterations required to reach the desired accuracy decreases as the mesh is refined.

Figure 1 depicts the errors $E\left(v\right)$ and $E\left(u\right)$ at the final time for $\Delta t=h^2$, $I=32$.

Next, we compare the efficiency of the proposed approach in (18), (20) with the standard second-order Crank–Nicolson method. Figure 2 presents the CPU time versus the accuracy in both the maximum and $L_2$ norms for each method at the final time with $\Delta t=h$. The results demonstrate that using the high-order scheme significantly reduces the computational time.

Example 2.

We consider the following test problem:

$$\begin{array}{cc}& {\displaystyle \tau =5,\phi \left(v\right)=v^4+2v,\psi \left(v\right)=2v^2+3,}\\ & {\displaystyle u_l\left(t\right)=\frac{\pi }{4}{e}^{t/2},u_r\left(t\right)=\frac{\pi \sqrt{2}}{8}{e}^{t/2},u^0\left(x\right)=sin\frac{\pi x}{4},}\\ & {\displaystyle f(x,t)=\frac{1}{2}{e}^{t/2}sin\frac{\pi x}{4}\left(1+\frac{{\pi }^2}{4}\left(1+\frac{{\pi }^3}{64}{e}^{3t/2}{cos}^3\frac{\pi x}{4}\right)+\frac{3{\pi }^3}{16}{e}^{t/2}cos\frac{\pi x}{4}\right).}\end{array}$$

In this case, the exact solution of the problem in (3), (9), (10) is $u(x,t)=e^{t/2}sin{\displaystyle \frac{\pi x}{4}}$.

In

Table 4. Errors and order of convergence of u, $\Delta t=h^2$, Example 2.

I	${\mathcal{E}}_{\mathit{\infty }}\left(\mathit{u}\right)$	${\mathcal{CR}}_{\mathit{\infty }}\left(\mathit{u}\right)$	${\mathcal{E}}_2\left(\mathit{u}\right)$	${\mathcal{CR}}_2\left(\mathit{u}\right)$	${\mathit{k}}_{\mathit{a}}$
8	8.8683 $\times {10}^{-6}$		5.0344 $\times {10}^{-6}$		3.000
16	9.3934 $\times {10}^{-7}$	3.2389	5.4452 $\times {10}^{-7}$	3.2088	3.000
32	7.0501 $\times {10}^{-8}$	3.7359	4.1561 $\times {10}^{-8}$	3.7117	3.000
64	4.8941 $\times {10}^{-9}$	3.8476	2.9230 $\times {10}^{-9}$	3.8297	3.000
128	3.2053 $\times {10}^{-10}$	3.9325	1.9107 $\times {10}^{-10}$	3.9353	2.000

, we present the errors and spatial order of convergence for solution u. The ratio $\Delta t=h^2$ was fixed for all runs. In

Table 5. Errors and order of convergence of u, $\Delta t=h$, Example 2.

N	${\mathcal{E}}_{\mathit{\infty }}\left(\mathit{u}\right)$	${\mathcal{CR}}_{\mathit{\infty }}\left(\mathit{u}\right)$	${\mathcal{E}}_2\left(\mathit{u}\right)$	${\mathcal{CR}}_2\left(\mathit{u}\right)$	${\mathit{k}}_{\mathit{a}}$
8	4.4115 $\times {10}^{-5}$		3.5683 $\times {10}^{-5}$		4.000
16	1.1347 $\times {10}^{-5}$	1.9590	9.1570 $\times {10}^{-6}$	1.9623	4.000
32	2.8750 $\times {10}^{-6}$	1.9807	2.3071 $\times {10}^{-6}$	1.9888	4.000
64	7.1814 $\times {10}^{-7}$	2.0012	5.7531 $\times {10}^{-7}$	2.0037	3.000
128	1.7889 $\times {10}^{-7}$	2.0052	1.4330 $\times {10}^{-7}$	2.0053	3.000

, we present computational results for numerical solution u when taking $\Delta t=h$. As in Example 1, we conclude that the accuracy of the proposed method is $O\left({(\Delta t)}^2+h^4\right)$.

In Figure 3, we plot the errors $E\left(v\right)$ and $E\left(u\right)$ in the whole computational domain for $\Delta t=h^2$, $I=32$.

Example 3.

Now, we set

$$\begin{array}{ccc}& & {\displaystyle \tau =0.5,\phi \left(v\right)=v,\psi \left(v\right)=sinv,u_l\left(t\right)=3t^4,u_r\left(t\right)=14t^4,u^0\left(x\right)=0,}\hfill \\ & & {\displaystyle f(x,t)=4t^4(x^5+2x^3+3x)-t^4(5x^4+6x^2+3)}\hfill \\ & & {\displaystyle -4t^4(20x^3+12)[cos(t^4(5x^4+6x^2+3)-t^4(5x^4+6x+3)sin\left(t^4(5x^4+6x^2+3)\right].}\hfill \end{array}$$

The exact solution to the problem in (3), (9), (10) is $u(x,t)=t^4(x^5+2x^3+3x)$.

Table 6. Errors and order of convergence of u, $\Delta t=h^2$, Example 3.

N	${\mathcal{E}}_{\mathit{\infty }}\left(\mathit{u}\right)$	${\mathcal{CR}}_{\mathit{\infty }}\left(\mathit{u}\right)$	${\mathcal{E}}_2\left(\mathit{u}\right)$	${\mathcal{CR}}_2\left(\mathit{u}\right)$	${\mathit{k}}_{\mathit{a}}$
8	9.1091 $\times {10}^{-4}$		6.1510 $\times {10}^{-4}$		3.129
16	4.3530 $\times {10}^{-5}$	4.3872	2.6453 $\times {10}^{-5}$	4.5393	2.921
32	2.5329 $\times {10}^{-6}$	4.1032	1.4890 $\times {10}^{-6}$	4.1510	2.683
64	1.5668 $\times {10}^{-7}$	4.0149	9.0914 $\times {10}^{-8}$	4.0337	2.579
128	9.4563 $\times {10}^{-9}$	4.0504	5.4044 $\times {10}^{-9}$	4.0723	2.444

and

Table 7. Errors and order of convergence of u, $\Delta t=h$, Example 3.

N	${\mathcal{E}}_{\mathit{\infty }}\left(\mathit{u}\right)$	${\mathcal{CR}}_{\mathit{\infty }}\left(\mathit{u}\right)$	${\mathcal{E}}_2\left(\mathit{u}\right)$	${\mathcal{CR}}_2\left(\mathit{u}\right)$	${\mathit{k}}_{\mathit{a}}$
8	6.8310 $\times {10}^{-2}$		5.3328 $\times {10}^{-2}$		4.000
16	1.0698 $\times {10}^{-2}$	2.6747	7.7448 $\times {10}^{-3}$	2.7836	3.429
32	2.2046 $\times {10}^{-3}$	2.2788	1.5678 $\times {10}^{-3}$	2.3048	3.267
64	5.1384 $\times {10}^{-4}$	2.1011	3.6202 $\times {10}^{-4}$	2.1143	3.129
128	1.2449 $\times {10}^{-4}$	2.0452	8.7290 $\times {10}^{-5}$	2.0522	3.048

present the results for numerical solutions v and u for $\Delta t=h^2$ and $\Delta t=h$, respectively. As in the previous two examples, we conclude that the accuracy of the developed approach is $O\left({(\Delta t)}^2+h^4\right)$.

Example 4.

In this example, we compare the solution gradient v computed using scheme (18) with those obtained in [2]. To verify their theoretical results, the authors of [2] employed an implicit finite difference scheme with accuracy $O(\Delta t+h^2)$ to solve the gradient problem in (11), (13) using a time step of $\Delta t=h$. Here, we adopt the same test example as in [2]:

$$\begin{array}{cc}& {\displaystyle \tau =0.5,\phi \left(v\right)=\frac{v}{1+v^2},\psi \left(v\right)=\frac{v^2}{1+v^2},f(x,t)=0,u_l\left(t\right)=u_r\left(t\right)=0,}\\ & {\displaystyle u^0\left(x\right)=4U_0{\left(\frac{4}{3}x\right)}^{7/3}(1-x).}\end{array}$$

In Figure 4, we compare the gradient of the solution obtained by both methods for different values of the constant $U_0$. Very good agreement is observed between the solution v computed by (18) and those computed in [2].

7. Conclusions

In this paper, we have proposed a finite difference method with fourth-order accuracy in space and second-order accuracy in time for numerically solving the initial Neumann boundary value problem for a pseudoparabolic equation. This equation models turbulent and mass-transfer diffusion in fluids, particularly the propagation of long waves in nonlinear dispersive systems. The original problem is reduced to a pseudoparabolic equation with an unknown gradient, subject to initial and Dirichlet boundary conditions. Our main contribution is the construction of a compact $O({\tau }^2+h^4)$ approximation for the reformulated problem. Newton’s iterative method is employed to solve the resulting nonlinear algebraic difference equations at each time step.

The computational results confirm that the order of convergence for both the solution and its derivative with respect to the spatial variable is fourth-order in space and second-order in time. A small number of iterations is required to achieve convergence.

The main advantages of the method developed in this work are as follows: first, both the solution and its gradient for the Barenblatt-type equation are computed with fourth-order accuracy in space and second-order in time; second, in contrast to existing studies, we address a more general problem involving a larger class of nonlinearities.

In future work, we intend to extend the proposed approach to pseudohyperbolic equations (see, e.g., [37]).

Moreover, an efficient approach based on backwards Taylor expansion for constructing a semi-implicit-type approximations was used in [42] to solve hyperbolic systems of balance laws. Applying this method to our problem would be another interesting direction for future research.