Volume 29, Issue 1
A Kernel Based Unconditionally Stable Scheme for Nonlinear Parabolic Partial Differential Equations

Kaipeng Wang, Andrew Christlieb, Yan JiangMengping Zhang

Commun. Comput. Phys., 29 (2021), pp. 237-264.

Published online: 2020-11

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  • Abstract

In this paper, a class of high order numerical schemes is proposed to solve the nonlinear parabolic equations with variable coefficients. This method is based on our previous work [11] for convection-diffusion equations, which relies on a special kernel-based formulation of the solutions and successive convolution. However, disadvantages appear when we extend the previous method to our equations, such as inefficient choice of parameters and unprovable stability for high-dimensional problems. To overcome these difficulties, a new kernel-based formulation is designed to approach the spatial derivatives. It maintains the good properties of the original one, including the high order accuracy and unconditionally stable for one-dimensional problems, hence allowing much larger time step evolution compared with other explicit schemes. In addition, without extra computational cost, the proposed scheme can enlarge the available interval of the special parameter in the formulation, leading to less errors and higher efficiency. Moreover, theoretical investigations indicate that it is unconditionally stable for multi-dimensional problems as well. We present numerical tests for one- and two-dimensional scalar and system, demonstrating the designed high order accuracy and unconditionally stable property of the scheme.

  • Keywords

Nonlinear parabolic equation, kernel based scheme, unconditionally stable, high order accuracy.

  • AMS Subject Headings

65M12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-29-237, author = {Wang , Kaipeng and Christlieb , Andrew and Jiang , Yan and Zhang , Mengping}, title = {A Kernel Based Unconditionally Stable Scheme for Nonlinear Parabolic Partial Differential Equations}, journal = {Communications in Computational Physics}, year = {2020}, volume = {29}, number = {1}, pages = {237--264}, abstract = {

In this paper, a class of high order numerical schemes is proposed to solve the nonlinear parabolic equations with variable coefficients. This method is based on our previous work [11] for convection-diffusion equations, which relies on a special kernel-based formulation of the solutions and successive convolution. However, disadvantages appear when we extend the previous method to our equations, such as inefficient choice of parameters and unprovable stability for high-dimensional problems. To overcome these difficulties, a new kernel-based formulation is designed to approach the spatial derivatives. It maintains the good properties of the original one, including the high order accuracy and unconditionally stable for one-dimensional problems, hence allowing much larger time step evolution compared with other explicit schemes. In addition, without extra computational cost, the proposed scheme can enlarge the available interval of the special parameter in the formulation, leading to less errors and higher efficiency. Moreover, theoretical investigations indicate that it is unconditionally stable for multi-dimensional problems as well. We present numerical tests for one- and two-dimensional scalar and system, demonstrating the designed high order accuracy and unconditionally stable property of the scheme.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2019-0197}, url = {http://global-sci.org/intro/article_detail/cicp/18429.html} }
TY - JOUR T1 - A Kernel Based Unconditionally Stable Scheme for Nonlinear Parabolic Partial Differential Equations AU - Wang , Kaipeng AU - Christlieb , Andrew AU - Jiang , Yan AU - Zhang , Mengping JO - Communications in Computational Physics VL - 1 SP - 237 EP - 264 PY - 2020 DA - 2020/11 SN - 29 DO - http://doi.org/10.4208/cicp.OA-2019-0197 UR - https://global-sci.org/intro/article_detail/cicp/18429.html KW - Nonlinear parabolic equation, kernel based scheme, unconditionally stable, high order accuracy. AB -

In this paper, a class of high order numerical schemes is proposed to solve the nonlinear parabolic equations with variable coefficients. This method is based on our previous work [11] for convection-diffusion equations, which relies on a special kernel-based formulation of the solutions and successive convolution. However, disadvantages appear when we extend the previous method to our equations, such as inefficient choice of parameters and unprovable stability for high-dimensional problems. To overcome these difficulties, a new kernel-based formulation is designed to approach the spatial derivatives. It maintains the good properties of the original one, including the high order accuracy and unconditionally stable for one-dimensional problems, hence allowing much larger time step evolution compared with other explicit schemes. In addition, without extra computational cost, the proposed scheme can enlarge the available interval of the special parameter in the formulation, leading to less errors and higher efficiency. Moreover, theoretical investigations indicate that it is unconditionally stable for multi-dimensional problems as well. We present numerical tests for one- and two-dimensional scalar and system, demonstrating the designed high order accuracy and unconditionally stable property of the scheme.

Kaipeng Wang, Andrew Christlieb, Yan Jiang & Mengping Zhang. (2020). A Kernel Based Unconditionally Stable Scheme for Nonlinear Parabolic Partial Differential Equations. Communications in Computational Physics. 29 (1). 237-264. doi:10.4208/cicp.OA-2019-0197
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