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Volume 42, Issue 1
Semi-Implicit Spectral Deferred Correction Methods Based on Second-Order Time Integration Schemes for Nonlinear PDEs

Ruihan Guo & Yan Xu

DOI: 10.4208/jcm.2202-m2021-0302

J. Comp. Math., 42 (2024), pp. 111-133.

Published online: 2023-12

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  • Abstract

In [20], a semi-implicit spectral deferred correction (SDC) method was proposed, which is efficient for highly nonlinear partial differential equations (PDEs). The semi-implicit SDC method in [20] is based on first-order time integration methods, which are corrected iteratively, with the order of accuracy increased by one for each additional iteration. In this paper, we will develop a class of semi-implicit SDC methods, which are based on second-order time integration methods and the order of accuracy are increased by two for each additional iteration. For spatial discretization, we employ the local discontinuous Galerkin (LDG) method to arrive at fully-discrete schemes, which are high-order accurate in both space and time. Numerical experiments are presented to demonstrate the accuracy, efficiency and robustness of the proposed semi-implicit SDC methods for solving complex nonlinear PDEs.

  • Keywords

Spectral deferred correction method, Nonlinear PDEs, Local discontinuous Galerkin method, Second-order scheme.

  • AMS Subject Headings

65M60, 35L75, 35G25

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-42-111, author = {Guo , Ruihan and Xu , Yan}, title = {Semi-Implicit Spectral Deferred Correction Methods Based on Second-Order Time Integration Schemes for Nonlinear PDEs}, journal = {Journal of Computational Mathematics}, year = {2023}, volume = {42}, number = {1}, pages = {111--133}, abstract = {

In [20], a semi-implicit spectral deferred correction (SDC) method was proposed, which is efficient for highly nonlinear partial differential equations (PDEs). The semi-implicit SDC method in [20] is based on first-order time integration methods, which are corrected iteratively, with the order of accuracy increased by one for each additional iteration. In this paper, we will develop a class of semi-implicit SDC methods, which are based on second-order time integration methods and the order of accuracy are increased by two for each additional iteration. For spatial discretization, we employ the local discontinuous Galerkin (LDG) method to arrive at fully-discrete schemes, which are high-order accurate in both space and time. Numerical experiments are presented to demonstrate the accuracy, efficiency and robustness of the proposed semi-implicit SDC methods for solving complex nonlinear PDEs.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2202-m2021-0302}, url = {http://global-sci.org/intro/article_detail/jcm/22154.html} }
TY - JOUR T1 - Semi-Implicit Spectral Deferred Correction Methods Based on Second-Order Time Integration Schemes for Nonlinear PDEs AU - Guo , Ruihan AU - Xu , Yan JO - Journal of Computational Mathematics VL - 1 SP - 111 EP - 133 PY - 2023 DA - 2023/12 SN - 42 DO - http://doi.org/10.4208/jcm.2202-m2021-0302 UR - https://global-sci.org/intro/article_detail/jcm/22154.html KW - Spectral deferred correction method, Nonlinear PDEs, Local discontinuous Galerkin method, Second-order scheme. AB -

In [20], a semi-implicit spectral deferred correction (SDC) method was proposed, which is efficient for highly nonlinear partial differential equations (PDEs). The semi-implicit SDC method in [20] is based on first-order time integration methods, which are corrected iteratively, with the order of accuracy increased by one for each additional iteration. In this paper, we will develop a class of semi-implicit SDC methods, which are based on second-order time integration methods and the order of accuracy are increased by two for each additional iteration. For spatial discretization, we employ the local discontinuous Galerkin (LDG) method to arrive at fully-discrete schemes, which are high-order accurate in both space and time. Numerical experiments are presented to demonstrate the accuracy, efficiency and robustness of the proposed semi-implicit SDC methods for solving complex nonlinear PDEs.

Guo , Ruihan and Xu , Yan. (2023). Semi-Implicit Spectral Deferred Correction Methods Based on Second-Order Time Integration Schemes for Nonlinear PDEs. Journal of Computational Mathematics. 42 (1). 111-133. doi:10.4208/jcm.2202-m2021-0302
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