**1. Introduction**

Most time-dependent differential equations are usually solved by multi-stage (one-step) method or multi-step method [1–3]. In general, there seems to be no significant difference in the structure between them when the multi-stage method is applied to ge<sup>t</sup> an initial guess for the multi-step method [4]. Nonetheless, a comparison of both methods has attracted quite a lot of interest from the viewpoints of convergence, stability, practical computations, numerical efficiency, etc. [5–11]. Comparisons in this regard do not take into account the impact of advances in computer science and technologies such as artificial intelligence (AI) or parallel computation, etc. Considering the impact, a new perspective to compare the potentials of both methods should be investigated as well as existing comparative studies. First of all, it is well known that the highest order of an *A*-stable multi-step method is two, so lots of research [12–24] developing higher order methods have focused on either multi-step methods satisfying some less restrictive stability condition or multi-stage methods which combine A-stability with high-order accuracy [2,25–29]. In addition, multi-stage methods such as Runge–Kutta (RK) type methods do not require any additional memory for function values at previous steps since it does not use any previously computed values [30–32]. On the other hand, multi-step methods require additional memory in the sense that they use previously computed function values and have insufficient function values for initial data. Multi-stage methods are comparable with multi-step methods for nonlinear stiff problems and have no restriction to express initial data contrast to the other. There seems not to be such a clear a priori distinction between multi-stage and multi-step methods.

Another interesting point of view to find more efficient methods is quite susceptible to stiffness and nonlinearity of the given problem. For nonlinear stiff problems, a multi-step method is needed to evaluate function values only once at each iteration in a nonlinear solver, whereas multi-stage methods require several function evaluations at each iteration. This disadvantage of the multi-stage method can be ignored by the authors' recent research [33]. The authors showed numerically that one stage of the multi-stage method is equivalent to one step of the multi-step method for simple ordinary differential equation (ODE) systems. However, the multi-step methods such as the backward differentiation formula (BDF) are usually recommended to apply nonlinear stiff problems because the process of solving the nonlinear system of equation is also expensive computationally. In the process of solving nonlinear stiff problems by a multi-stage method, it generally generates a system *Md* ⊗ *Ms*, where *d* and *s* represent the dimension of the given problem and the number of stages used in the multi-stage method, respectively. Here, the notation *Mk* represents a matrix with the size *k* × *k* and the notation ⊗ denotes a Kronecker product. On the other hand, a multi-step method needs to solve only a system of size *d* × *d*.

The purpose of this paper is to investigate and compare the properties of the multi-stage and the multi-step methods for *d*-dimensional stiff problems described by

$$\frac{dy}{dt} = f(t, y) \in \mathbb{R}^d. \tag{1}$$

Most nonlinear stiff problems are solved by multi-step methods rather than multi-stage methods since the multi-stage methods usually transform nonlinear stiff problems into bigger nonlinear systems, as mentioned in the previous paragraph. To solve such nonlinear systems efficiently, one has to consider both nonlinear and linear solvers. The nonlinear systems are usually solved by using an iteration technique such as Newton-like iterations, which incur considerable computation costs. There are various Newton-like iterations. Among them, a simplified Newton iteration is developed in connection with the development of computer process capacity [34–37]. Different nonlinear system solvers generate linear systems correspondingly. It means that the nonlinear system solver should be well-selected to adapt efficient linear solvers such as the eigenvalue decomposition method. Note that efficient linear solvers have also been well-studied [1,2,38,39]. An eigenvalue decomposition combined with simplified Newton iteration can apply to a multi-stage method. The resulting multi-stage method generates the same matrix, regardless of integration or iteration, as an object of decomposition for solving a linear system induced by the simplified Newton iteration. It allows for decomposing the matrix only once throughout the whole process. As a result, applying this combination to multi-stage methods highlights the advantage of multi-stage methods by reducing computational costs to the level of the costs required from multi-step methods without any loss of the original advantages of multi-stage methods, which is the main contention of this paper.

The remaining parts of this paper are as follows. We briefly describe the multi-step and multi-stage methods and simplified Newton iteration in Section 2. To support theoretical analysis, we present preliminary numerical results in Section 3. Finally, in Section 4, all results are summarized and further possibilities are discussed.
