Volume 26, Issue 1
A Novel Method for Solving Time-Dependent 2D Advection-Diffusion-Reaction Equations to Model Transfer in Nonlinear Anisotropic Media

Ji Lin, Sergiy Reutskiy, C. S. Chen & Jun Lu

Commun. Comput. Phys., 26 (2019), pp. 233-264.

Published online: 2019-02

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  • Abstract

This paper presents a new numerical technique for solving initial and boundary value problems with unsteady strongly nonlinear advection diffusion reaction (ADR) equations. The method is based on the use of the radial basis functions (RBF) for the approximation space of the solution. The Crank-Nicolson scheme is used for approximation in time. This results in a sequence of stationary nonlinear ADR equations. The equations are solved sequentially at each time step using the proposed semi-analytical technique based on the RBFs. The approximate solution is sought in the form of the analytical expansion over basis functions and contains free parameters. The basis functions are constructed in such a way that the expansion satisfies the boundary conditions of the problem for any choice of the free parameters. The free parameters are determined by substitution of the expansion in the equation and collocation in the solution domain. In the case of a nonlinear equation, we use the well-known procedure of quasilinearization. This transforms the original equation into a sequence of the linear ones on each time layer. The numerical examples confirm the high accuracy and robustness of the proposed numerical scheme.

  • Keywords

Advection diffusion reaction, time-dependent, fully nonlinear, anisotropic media, Crank-Nicolson scheme, meshless method.

  • AMS Subject Headings

65N35, 65N40, 65Y20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-26-233, author = {}, title = {A Novel Method for Solving Time-Dependent 2D Advection-Diffusion-Reaction Equations to Model Transfer in Nonlinear Anisotropic Media}, journal = {Communications in Computational Physics}, year = {2019}, volume = {26}, number = {1}, pages = {233--264}, abstract = {

This paper presents a new numerical technique for solving initial and boundary value problems with unsteady strongly nonlinear advection diffusion reaction (ADR) equations. The method is based on the use of the radial basis functions (RBF) for the approximation space of the solution. The Crank-Nicolson scheme is used for approximation in time. This results in a sequence of stationary nonlinear ADR equations. The equations are solved sequentially at each time step using the proposed semi-analytical technique based on the RBFs. The approximate solution is sought in the form of the analytical expansion over basis functions and contains free parameters. The basis functions are constructed in such a way that the expansion satisfies the boundary conditions of the problem for any choice of the free parameters. The free parameters are determined by substitution of the expansion in the equation and collocation in the solution domain. In the case of a nonlinear equation, we use the well-known procedure of quasilinearization. This transforms the original equation into a sequence of the linear ones on each time layer. The numerical examples confirm the high accuracy and robustness of the proposed numerical scheme.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2018-0005}, url = {http://global-sci.org/intro/article_detail/cicp/13033.html} }
TY - JOUR T1 - A Novel Method for Solving Time-Dependent 2D Advection-Diffusion-Reaction Equations to Model Transfer in Nonlinear Anisotropic Media JO - Communications in Computational Physics VL - 1 SP - 233 EP - 264 PY - 2019 DA - 2019/02 SN - 26 DO - http://dor.org/10.4208/cicp.OA-2018-0005 UR - https://global-sci.org/intro/article_detail/cicp/13033.html KW - Advection diffusion reaction, time-dependent, fully nonlinear, anisotropic media, Crank-Nicolson scheme, meshless method. AB -

This paper presents a new numerical technique for solving initial and boundary value problems with unsteady strongly nonlinear advection diffusion reaction (ADR) equations. The method is based on the use of the radial basis functions (RBF) for the approximation space of the solution. The Crank-Nicolson scheme is used for approximation in time. This results in a sequence of stationary nonlinear ADR equations. The equations are solved sequentially at each time step using the proposed semi-analytical technique based on the RBFs. The approximate solution is sought in the form of the analytical expansion over basis functions and contains free parameters. The basis functions are constructed in such a way that the expansion satisfies the boundary conditions of the problem for any choice of the free parameters. The free parameters are determined by substitution of the expansion in the equation and collocation in the solution domain. In the case of a nonlinear equation, we use the well-known procedure of quasilinearization. This transforms the original equation into a sequence of the linear ones on each time layer. The numerical examples confirm the high accuracy and robustness of the proposed numerical scheme.

Ji Lin, Sergiy Reutskiy, C. S. Chen & Jun Lu. (2019). A Novel Method for Solving Time-Dependent 2D Advection-Diffusion-Reaction Equations to Model Transfer in Nonlinear Anisotropic Media. Communications in Computational Physics. 26 (1). 233-264. doi:10.4208/cicp.OA-2018-0005
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