Figure 1.
Plot of the exact RPE vs. k vs. for the three methods using . (a) Kowalic–Murty with ; (b) Lax–Wendroff with ; (c) NSFD with .
Figure 1.
Plot of the exact RPE vs. k vs. for the three methods using . (a) Kowalic–Murty with ; (b) Lax–Wendroff with ; (c) NSFD with .
Figure 2.
Plot of integrated error vs. k to determine optimal k when . (a) Lax–Wendroff scheme; (b) NSFD scheme.
Figure 2.
Plot of integrated error vs. k to determine optimal k when . (a) Lax–Wendroff scheme; (b) NSFD scheme.
Figure 3.
Plot of the exact RPE vs. for the three methods at different values of k when . (a) Kowalic–Murty. (b) Lax–Wendroff. (c) NSFD.
Figure 3.
Plot of the exact RPE vs. for the three methods at different values of k when . (a) Kowalic–Murty. (b) Lax–Wendroff. (c) NSFD.
Figure 4.
Plots of numerical and exact solution vs. x at time when = 0.005 m/s and U = 0.8 m/s using the three schemes at and different values of k. (a) Initial and Exact; (b) Kowalic–Murty; (c) Lax–Wendroff; (d) NSFD.
Figure 4.
Plots of numerical and exact solution vs. x at time when = 0.005 m/s and U = 0.8 m/s using the three schemes at and different values of k. (a) Initial and Exact; (b) Kowalic–Murty; (c) Lax–Wendroff; (d) NSFD.
Figure 5.
Plot of absolute errors vs. x using the three methods at and different values of k. (a) Kowalic–Murty scheme with ; (b) Kowalic–Murty scheme with ; (c) Lax–Wendroff scheme with ; (d) Lax–Wendroff scheme with ; (e) NSFD scheme with ; (f) NSFD scheme with .
Figure 5.
Plot of absolute errors vs. x using the three methods at and different values of k. (a) Kowalic–Murty scheme with ; (b) Kowalic–Murty scheme with ; (c) Lax–Wendroff scheme with ; (d) Lax–Wendroff scheme with ; (e) NSFD scheme with ; (f) NSFD scheme with .
Figure 6.
Plots of the initial profile, numerical, and exact solutions at time and when = 0.8 m/s and U = 0.005 m/s using the three schemes at different values of k. (a) ; (b) ; (c) ; (d) ; (e) ; (f) .
Figure 6.
Plots of the initial profile, numerical, and exact solutions at time and when = 0.8 m/s and U = 0.005 m/s using the three schemes at different values of k. (a) ; (b) ; (c) ; (d) ; (e) ; (f) .
Figure 7.
The 3D plot of solution vs. x vs. y at time with , , . (a) Initial profile; (b) exact profile.
Figure 7.
The 3D plot of solution vs. x vs. y at time with , , . (a) Initial profile; (b) exact profile.
Figure 8.
The 3D plot of the solution using the Kowalic–Murty scheme vs. x vs. y at time with , , , . (a) ; (b) ; (c) .
Figure 8.
The 3D plot of the solution using the Kowalic–Murty scheme vs. x vs. y at time with , , , . (a) ; (b) ; (c) .
Figure 9.
The 3D plot of the solution using the Lax–Wendroff scheme vs. x vs. y at time with , , , . (a) ; (b) ; (c) .
Figure 9.
The 3D plot of the solution using the Lax–Wendroff scheme vs. x vs. y at time with , , , . (a) ; (b) ; (c) .
Figure 10.
The 3D plot of the solution using the NSFD scheme vs. x vs. y at time with , , , . (a) ; (b) ; (c) .
Figure 10.
The 3D plot of the solution using the NSFD scheme vs. x vs. y at time with , , , . (a) ; (b) ; (c) .
Figure 11.
Plot of numerical solution vs. x vs. y at time when , , , , on using the scheme from the Kowalic–Murty scheme. (a) Initial profile; (b) exact solution; (c) numerical profile at ; (d) numerical profile at ; (e) numerical profile at .
Figure 11.
Plot of numerical solution vs. x vs. y at time when , , , , on using the scheme from the Kowalic–Murty scheme. (a) Initial profile; (b) exact solution; (c) numerical profile at ; (d) numerical profile at ; (e) numerical profile at .
Figure 12.
Plot of the numerical solution vs. x vs. y at time when , , , , on using Lax–Wendroff. (a) Initial profile; (b) exact solution; (c) numerical profile at ; (d) numerical profile at ; (e) numerical profile at .
Figure 12.
Plot of the numerical solution vs. x vs. y at time when , , , , on using Lax–Wendroff. (a) Initial profile; (b) exact solution; (c) numerical profile at ; (d) numerical profile at ; (e) numerical profile at .
Figure 13.
Plot of the numerical solution vs. x vs. y at time when , , , , on using NSFD. (a) Initial profile; (b) exact solution; (c) numerical profile at ; (d) numerical profile at ; (e) numerical profile at .
Figure 13.
Plot of the numerical solution vs. x vs. y at time when , , , , on using NSFD. (a) Initial profile; (b) exact solution; (c) numerical profile at ; (d) numerical profile at ; (e) numerical profile at .
Table 1.
, errors, and total mean square error at time with different values of k using Kowalic–Murty, Lax–Wendroff, and NSFD schemes using for numerical experiment 1.
Table 1.
, errors, and total mean square error at time with different values of k using Kowalic–Murty, Lax–Wendroff, and NSFD schemes using for numerical experiment 1.
Schemes | k | Error | Error | TMSE |
---|
Kowalic–Murty | 0.0125 | | | |
| 0.0050 | | | |
| 0.0025 | | | |
| 0.00125 | | | |
Lax–Wendroff | 0.0200 | | | |
| 0.015625 | | | |
| 0.0100 | | | |
| 0.0050 | | | |
NSFD | 0.0200 | | | |
| 0.0149518 | | | |
| 0.0100 | | | |
| 0.0050 | | | |
Table 2.
Rate of convergence in space for the three schemes when used to solve numerical experiment 1 at time .
Table 2.
Rate of convergence in space for the three schemes when used to solve numerical experiment 1 at time .
Schemes | h | Error | Error | Rate of Convergence in Space |
---|
Kowalic–Murty | 0.0500 | 0.0471 | 0.0865 | – |
| 0.0250 | 0.0096 | 0.0174 | 2.2946 |
| 0.0125 | 0.0023 | 0.0042 | 2.0614 |
| 0.00625 | | 0,0010 | 1.9925 |
Lax–Wendroff | 0.1000 | 0.0609 | 0.0868 | – |
| 0.0500 | 0.0221 | 0.0333 | 1.5992 |
| 0.0250 | 0.0043 | 0.0070 | 2.2248 |
| 0.0125 | | | 2.8554 |
| 0.00625 | | | 2.5859 |
NSFD | 0.1000 | 0.0814 | 0.1290 | – |
| 0.0500 | 0.0576 | 0.0943 | 0.4990 |
| 0.0250 | 0.0267 | 0.0460 | 1.1092 |
Table 3.
Rate of convergence in space using the three schemes for numerical experiment 2 at time .
Table 3.
Rate of convergence in space using the three schemes for numerical experiment 2 at time .
Schemes | h | Error | Error | Rate of Convergence in Space |
---|
Kowalic–Murty scheme [11] | 0.1000 | | | – |
| 0.0500 | | | 2.0001 |
| 0.0250 | | | 2.0001 |
| 0.0125 | | | 2.0001 |
Lax–Wendroff | 0.1000 | | | – |
| 0.0500 | | | 1.9998 |
| 0.0250 | | | 1.9999 |
| 0.0125 | | | 2.0000 |
NSFD | 0.1000 | 0.0065 | 0.0032 | – |
| 0.0500 | 0.0032 | 0.0016 | 1.0224 |
| 0.0250 | 0.0016 | | 1.0000 |
| 0.0125 | | | 1.0023 |
Table 4.
Stability region of the three methods discretizing 2D advection–diffusion equation for numerical experiments 3 and 4.
Table 4.
Stability region of the three methods discretizing 2D advection–diffusion equation for numerical experiments 3 and 4.
Schemes | Numerical Experiment | Value of h | Stability Region |
---|
Kowalic–Murty | 3 | | |
| 4 | 5000 | |
Lax–Wendroff | 3 | | |
| 4 | 5000 | |
NSFD | 3 | 0.025 | |
| 4 | 5000 | |
Table 5.
, errors, and rate of convergence in time from the three methods for , at time using and different values of k.
Table 5.
, errors, and rate of convergence in time from the three methods for , at time using and different values of k.
Schemes | Value of k | Error | Error | Rate of Convergence in Time |
---|
Kowalic–Murty scheme [11] | 0.008 | | | – |
| | | | 1.3219 |
| | 0.0015 | 0.0031 | 1.1806 |
Lax–Wendroff | 0.008 | | | – |
| 0.004 | | | 1.0418 |
| | | | 0.8021 |
NSFD | 0.008 | 0.0038 | | – |
| 0.004 | 0.0022 | | 0.7885 |
Table 6.
and errors at time using the three methods for , , at and different values of k.
Table 6.
and errors at time using the three methods for , , at and different values of k.
Schemes | Value of k | Error | Error |
---|
Kowalic–Murty scheme [11] | 300 | | |
| 150 | | 14.0042 |
Lax–Wendroff | 600 | | |
| 300 | | |
| 150 | | 1.4263 |
NSFD | 600 | | |
| 300 | | |
| 150 | | 27.5904 |