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Volume 35, Issue 2
Maximum-Principle-Preserving, Steady-State-Preserving and Large Time-Stepping High-Order Schemes for Scalar Hyperbolic Equations with Source Terms

Lele Liu, Hong Zhang, Xu Qian & Songhe Song

Commun. Comput. Phys., 35 (2024), pp. 498-523.

Published online: 2024-03

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  • Abstract

In this paper, we construct a family of temporal high-order parametric relaxation Runge–Kutta (pRRK) schemes for stiff ordinary differential equations (ODEs), and explore their application in solving hyperbolic conservation laws with source terms. The new time discretization methods are explicit, large time-stepping, delay-free and able to preserve steady state. They are combined with fifth-order weighted compact nonlinear schemes (WCNS5) spatial discretization and parametrized maximum-principle-preserving (MPP) flux limiters to solve scalar hyperbolic equations with source terms. We prove that the fully discrete schemes preserve the maximum principle strictly. Through benchmark test problems, we demonstrate that the proposed schemes have fifth-order accuracy in space, fourth-order accuracy in time and allow for large time-stepping without time delay. Both theoretical analyses and numerical experiments are presented to validate the benefits of the proposed schemes.

  • AMS Subject Headings

34D20, 35L65, 65M06, 65M20, 76M20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-35-498, author = {Liu , LeleZhang , HongQian , Xu and Song , Songhe}, title = {Maximum-Principle-Preserving, Steady-State-Preserving and Large Time-Stepping High-Order Schemes for Scalar Hyperbolic Equations with Source Terms}, journal = {Communications in Computational Physics}, year = {2024}, volume = {35}, number = {2}, pages = {498--523}, abstract = {

In this paper, we construct a family of temporal high-order parametric relaxation Runge–Kutta (pRRK) schemes for stiff ordinary differential equations (ODEs), and explore their application in solving hyperbolic conservation laws with source terms. The new time discretization methods are explicit, large time-stepping, delay-free and able to preserve steady state. They are combined with fifth-order weighted compact nonlinear schemes (WCNS5) spatial discretization and parametrized maximum-principle-preserving (MPP) flux limiters to solve scalar hyperbolic equations with source terms. We prove that the fully discrete schemes preserve the maximum principle strictly. Through benchmark test problems, we demonstrate that the proposed schemes have fifth-order accuracy in space, fourth-order accuracy in time and allow for large time-stepping without time delay. Both theoretical analyses and numerical experiments are presented to validate the benefits of the proposed schemes.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2023-0143}, url = {http://global-sci.org/intro/article_detail/cicp/22980.html} }
TY - JOUR T1 - Maximum-Principle-Preserving, Steady-State-Preserving and Large Time-Stepping High-Order Schemes for Scalar Hyperbolic Equations with Source Terms AU - Liu , Lele AU - Zhang , Hong AU - Qian , Xu AU - Song , Songhe JO - Communications in Computational Physics VL - 2 SP - 498 EP - 523 PY - 2024 DA - 2024/03 SN - 35 DO - http://doi.org/10.4208/cicp.OA-2023-0143 UR - https://global-sci.org/intro/article_detail/cicp/22980.html KW - Maximum-principle-preserving, steady-state-preserving, parametric relaxation Runge–Kutta schemes, weighted compact nonlinear schemes, hyperbolic equations. AB -

In this paper, we construct a family of temporal high-order parametric relaxation Runge–Kutta (pRRK) schemes for stiff ordinary differential equations (ODEs), and explore their application in solving hyperbolic conservation laws with source terms. The new time discretization methods are explicit, large time-stepping, delay-free and able to preserve steady state. They are combined with fifth-order weighted compact nonlinear schemes (WCNS5) spatial discretization and parametrized maximum-principle-preserving (MPP) flux limiters to solve scalar hyperbolic equations with source terms. We prove that the fully discrete schemes preserve the maximum principle strictly. Through benchmark test problems, we demonstrate that the proposed schemes have fifth-order accuracy in space, fourth-order accuracy in time and allow for large time-stepping without time delay. Both theoretical analyses and numerical experiments are presented to validate the benefits of the proposed schemes.

Lele Liu, Hong Zhang, Xu Qian & Songhe Song. (2024). Maximum-Principle-Preserving, Steady-State-Preserving and Large Time-Stepping High-Order Schemes for Scalar Hyperbolic Equations with Source Terms. Communications in Computational Physics. 35 (2). 498-523. doi:10.4208/cicp.OA-2023-0143
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