Linearized Crank–Nicolson Scheme for the Two-Dimensional Nonlinear Riesz Space-Fractional Convection–Diffusion Equation
Abstract
:1. Introduction
2. Preliminaries
3. CNADI-WSGD Method
- (1)
- To obtain the intermediate solution , we solve a set of equations that define by (19) for every fixed at every mesh point .
- (2)
- To find the numerical solution , we solve a set of equations that define by (20) for each fixed at the points by alternating the spatial direction.
- (3)
- The homogeneous Dirichlet boundary conditions are used as the following:
4. Theoretical Analysis of the CNADI–WSGD Scheme
4.1. Stability of the CNADI–WSGD Approximation
4.2. Convergence of the CNADI–WSGD Approximation
5. Numerical Examples
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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order | order | CPU(s) | ||||
---|---|---|---|---|---|---|
(1.2, 1.8) | 1/4 | - | - | 0.0020 | ||
1/8 | 1.7962 | 1.8984 | 0.0059 | |||
1/16 | 1.8128 | 2.0041 | 0.0288 | |||
1/32 | 2.0094 | 2.0154 | 0.1724 | |||
1/64 | 1.9945 | 2.0077 | 0.8077 | |||
1/128 | 2.0013 | 2.0006 | 5.8330 | |||
(1.3, 1.7) | 1/4 | - | - | 0.0022 | ||
1/8 | 1.7694 | 1.8583 | 0.0100 | |||
1/16 | 1.834 | 1.9669 | 0.0837 | |||
1/32 | 1.9703 | 1.985 | 0.1850 | |||
1/64 | 1.9805 | 1.9903 | 1.0993 | |||
1/128 | 1.9949 | 1.9932 | 8.0243 | |||
(1.4, 1.6) | 1/4 | - | - | 0.0020 | ||
1/8 | 1.7535 | 1.8322 | 0.0066 | |||
1/16 | 1.8615 | 1.9434 | 0.0355 | |||
1/32 | 1.923 | 1.9669 | 0.1485 | |||
1/64 | 1.9806 | 1.9806 | 0.8680 | |||
1/128 | 1.991 | 1.989 | 6.9709 |
CN–ADI | CN–non–ADI | ||||||
---|---|---|---|---|---|---|---|
Order | CPU | Order | CPU | ||||
1/40 | - | 0.1268 | - | 0.2446 | |||
1/80 | 2.0025 | 0.7686 | 2.0025 | 1.3055 | |||
(1.2,1.8) | 1/160 | 2.0004 | 5.9595 | 2.0004 | 9.8197 | ||
1/320 | 1.9994 | 61.1540 | 1.9994 | 72.0935 | |||
1/40 | - | 0.1395 | - | 0.2249 | |||
1/80 | 1.9939 | 1.1048 | 1.9939 | 1.4776 | |||
(1.3,1.7) | 1/160 | 1.995 | 7.7145 | 1.995 | 11.2881 | ||
1/320 | 1.9978 | 62.0667 | 1.9978 | 82.0642 | |||
1/40 | - | 0.1492 | - | 0.2446 | |||
1/80 | 1.9829 | 1.0746 | 1.9829 | 1.3055 | |||
(1.4,1.6) | 1/160 | 1.9938 | 7.7024 | 1.9938 | 9.8197 | ||
1/320 | 1.997 | 61.3860 | 1.997 | 72.0935 |
order | order | CPU(s) | ||||
---|---|---|---|---|---|---|
(1.2, 1.8) | 1/4 | - | - | 0.0164 | ||
1/8 | 2.0527 | 2.0099 | 0.0076 | |||
1/16 | 2.0303 | 2.0295 | 0.0155 | |||
1/32 | 2.0308 | 2.0431 | 0.1361 | |||
1/64 | 2.0335 | 2.0479 | 0.6257 | |||
1/128 | 1.842 | 2.0467 | 3.8980 | |||
(1.3, 1.7) | 1/4 | - | - | 0.0077 | ||
1/8 | 2.0566 | 2.0263 | 0.0077 | |||
1/16 | 2.0378 | 2.0394 | 0.0336 | |||
1/32 | 2.0429 | 2.0442 | 0.1148 | |||
1/64 | 1.9079 | 2.0406 | 0.5898 | |||
1/128 | 1.715 | 2.033 | 3.9085 | |||
(1.4, 1.6) | 1/4 | - | - | 0.0038 | ||
1/8 | 2.0599 | 2.0349 | 0.0084 | |||
1/16 | 2.0428 | 2.0438 | 0.0367 | |||
1/32 | 2.0509 | 2.0419 | 0.1276 | |||
1/64 | 1.9846 | 2.0302 | 0.5947 | |||
1/128 | 1.7132 | 2.0154 | 3.9343 |
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Basha, M.; Anley, E.F.; Dai, B. Linearized Crank–Nicolson Scheme for the Two-Dimensional Nonlinear Riesz Space-Fractional Convection–Diffusion Equation. Fractal Fract. 2023, 7, 240. https://doi.org/10.3390/fractalfract7030240
Basha M, Anley EF, Dai B. Linearized Crank–Nicolson Scheme for the Two-Dimensional Nonlinear Riesz Space-Fractional Convection–Diffusion Equation. Fractal and Fractional. 2023; 7(3):240. https://doi.org/10.3390/fractalfract7030240
Chicago/Turabian StyleBasha, Merfat, Eyaya Fekadie Anley, and Binxiang Dai. 2023. "Linearized Crank–Nicolson Scheme for the Two-Dimensional Nonlinear Riesz Space-Fractional Convection–Diffusion Equation" Fractal and Fractional 7, no. 3: 240. https://doi.org/10.3390/fractalfract7030240
APA StyleBasha, M., Anley, E. F., & Dai, B. (2023). Linearized Crank–Nicolson Scheme for the Two-Dimensional Nonlinear Riesz Space-Fractional Convection–Diffusion Equation. Fractal and Fractional, 7(3), 240. https://doi.org/10.3390/fractalfract7030240