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Volume 30, Issue 1
Adaptive Order WENO Reconstructions for the Semi-Lagrangian Finite Difference Scheme for Advection Problem

Jiajie Chen, Xiaofeng Cai, Jianxian Qiu & Jing-Mei Qiu

DOI: 10.4208/cicp.OA-2020-0073

Commun. Comput. Phys., 30 (2021), pp. 67-96.

Published online: 2021-04

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  • Abstract

We present a new conservative semi-Lagrangian finite difference weighted essentially non-oscillatory scheme with adaptive order. This is an extension of the conservative semi-Lagrangian (SL) finite difference WENO scheme in [Qiu and Shu, JCP, 230 (4) (2011), pp. 863-889], in which linear weights in SL WENO framework were shown not to exist for variable coefficient problems. Hence, the order of accuracy is not optimal from reconstruction stencils. In this paper, we incorporate a recent WENO adaptive order (AO) technique [Balsara et al., JCP, 326 (2016), pp. 780-804] to the SL WENO framework. The new scheme can achieve an optimal high order of accuracy, while maintaining the properties of mass conservation and non-oscillatory capture of solutions from the original SL WENO. The positivity-preserving limiter is further applied to ensure the positivity of solutions. Finally, the scheme is applied to high dimensional problems by a fourth-order dimensional splitting. We demonstrate the effectiveness of the new scheme by extensive numerical tests on linear advection equations, the Vlasov-Poisson system, the guiding center Vlasov model as well as the incompressible Euler equations.

  • Keywords

Semi-Lagrangian, weighted essentially non-oscillatory, WENO adaptive order reconstruction, finite difference, mass conservation, Vlasov-Poisson, incompressible Euler.

  • AMS Subject Headings

65

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COPYRIGHT: © Global Science Press

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@Article{CiCP-30-67, author = {Chen , JiajieCai , XiaofengQiu , Jianxian and Qiu , Jing-Mei}, title = {Adaptive Order WENO Reconstructions for the Semi-Lagrangian Finite Difference Scheme for Advection Problem}, journal = {Communications in Computational Physics}, year = {2021}, volume = {30}, number = {1}, pages = {67--96}, abstract = {

We present a new conservative semi-Lagrangian finite difference weighted essentially non-oscillatory scheme with adaptive order. This is an extension of the conservative semi-Lagrangian (SL) finite difference WENO scheme in [Qiu and Shu, JCP, 230 (4) (2011), pp. 863-889], in which linear weights in SL WENO framework were shown not to exist for variable coefficient problems. Hence, the order of accuracy is not optimal from reconstruction stencils. In this paper, we incorporate a recent WENO adaptive order (AO) technique [Balsara et al., JCP, 326 (2016), pp. 780-804] to the SL WENO framework. The new scheme can achieve an optimal high order of accuracy, while maintaining the properties of mass conservation and non-oscillatory capture of solutions from the original SL WENO. The positivity-preserving limiter is further applied to ensure the positivity of solutions. Finally, the scheme is applied to high dimensional problems by a fourth-order dimensional splitting. We demonstrate the effectiveness of the new scheme by extensive numerical tests on linear advection equations, the Vlasov-Poisson system, the guiding center Vlasov model as well as the incompressible Euler equations.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2020-0073}, url = {http://global-sci.org/intro/article_detail/cicp/18874.html} }
TY - JOUR T1 - Adaptive Order WENO Reconstructions for the Semi-Lagrangian Finite Difference Scheme for Advection Problem AU - Chen , Jiajie AU - Cai , Xiaofeng AU - Qiu , Jianxian AU - Qiu , Jing-Mei JO - Communications in Computational Physics VL - 1 SP - 67 EP - 96 PY - 2021 DA - 2021/04 SN - 30 DO - http://doi.org/10.4208/cicp.OA-2020-0073 UR - https://global-sci.org/intro/article_detail/cicp/18874.html KW - Semi-Lagrangian, weighted essentially non-oscillatory, WENO adaptive order reconstruction, finite difference, mass conservation, Vlasov-Poisson, incompressible Euler. AB -

We present a new conservative semi-Lagrangian finite difference weighted essentially non-oscillatory scheme with adaptive order. This is an extension of the conservative semi-Lagrangian (SL) finite difference WENO scheme in [Qiu and Shu, JCP, 230 (4) (2011), pp. 863-889], in which linear weights in SL WENO framework were shown not to exist for variable coefficient problems. Hence, the order of accuracy is not optimal from reconstruction stencils. In this paper, we incorporate a recent WENO adaptive order (AO) technique [Balsara et al., JCP, 326 (2016), pp. 780-804] to the SL WENO framework. The new scheme can achieve an optimal high order of accuracy, while maintaining the properties of mass conservation and non-oscillatory capture of solutions from the original SL WENO. The positivity-preserving limiter is further applied to ensure the positivity of solutions. Finally, the scheme is applied to high dimensional problems by a fourth-order dimensional splitting. We demonstrate the effectiveness of the new scheme by extensive numerical tests on linear advection equations, the Vlasov-Poisson system, the guiding center Vlasov model as well as the incompressible Euler equations.

Jiajie Chen, Xiaofeng Cai, Jianxian Qiu & Jing-Mei Qiu. (2021). Adaptive Order WENO Reconstructions for the Semi-Lagrangian Finite Difference Scheme for Advection Problem. Communications in Computational Physics. 30 (1). 67-96. doi:10.4208/cicp.OA-2020-0073
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