Krylov SSP Integrating Factor Runge–Kutta WENO Methods
Weighted essentially non-oscillatory (WENO) methods are especially efficient for numerically solving nonlinear hyperbolic equations. In order to achieve strong stability and large time-steps, strong stability preserving (SSP) integrating factor (IF) methods were designed in the literature, but the m...
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| Format: | Article |
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MDPI AG
2021-06-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/13/1483 |
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| author | Shanqin Chen |
| author_facet | Shanqin Chen |
| author_sort | Shanqin Chen |
| collection | DOAJ |
| description | Weighted essentially non-oscillatory (WENO) methods are especially efficient for numerically solving nonlinear hyperbolic equations. In order to achieve strong stability and large time-steps, strong stability preserving (SSP) integrating factor (IF) methods were designed in the literature, but the methods there were only for one-dimensional (1D) problems that have a stiff linear component and a non-stiff nonlinear component. In this paper, we extend WENO methods with large time-stepping SSP integrating factor Runge–Kutta time discretization to solve general nonlinear two-dimensional (2D) problems by a splitting method. How to evaluate the matrix exponential operator efficiently is a tremendous challenge when we apply IF temporal discretization for PDEs on high spatial dimensions. In this work, the matrix exponential computation is approximated through the Krylov subspace projection method. Numerical examples are shown to demonstrate the accuracy and large time-step size of the present method. |
| first_indexed | 2024-03-10T10:06:04Z |
| format | Article |
| id | doaj.art-f4540e7e410642d390a49cd895363865 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T10:06:04Z |
| publishDate | 2021-06-01 |
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| series | Mathematics |
| spelling | doaj.art-f4540e7e410642d390a49cd8953638652023-11-22T01:33:16ZengMDPI AGMathematics2227-73902021-06-01913148310.3390/math9131483Krylov SSP Integrating Factor Runge–Kutta WENO MethodsShanqin Chen0Department of Mathematical Sciences, Indiana University South Bend, South Bend, IN 46615, USAWeighted essentially non-oscillatory (WENO) methods are especially efficient for numerically solving nonlinear hyperbolic equations. In order to achieve strong stability and large time-steps, strong stability preserving (SSP) integrating factor (IF) methods were designed in the literature, but the methods there were only for one-dimensional (1D) problems that have a stiff linear component and a non-stiff nonlinear component. In this paper, we extend WENO methods with large time-stepping SSP integrating factor Runge–Kutta time discretization to solve general nonlinear two-dimensional (2D) problems by a splitting method. How to evaluate the matrix exponential operator efficiently is a tremendous challenge when we apply IF temporal discretization for PDEs on high spatial dimensions. In this work, the matrix exponential computation is approximated through the Krylov subspace projection method. Numerical examples are shown to demonstrate the accuracy and large time-step size of the present method.https://www.mdpi.com/2227-7390/9/13/1483strong stability preservingintegrating factorRunge–Kuttaweighted essentially non-oscillatory methodsKrylov subspace approximation |
| spellingShingle | Shanqin Chen Krylov SSP Integrating Factor Runge–Kutta WENO Methods Mathematics strong stability preserving integrating factor Runge–Kutta weighted essentially non-oscillatory methods Krylov subspace approximation |
| title | Krylov SSP Integrating Factor Runge–Kutta WENO Methods |
| title_full | Krylov SSP Integrating Factor Runge–Kutta WENO Methods |
| title_fullStr | Krylov SSP Integrating Factor Runge–Kutta WENO Methods |
| title_full_unstemmed | Krylov SSP Integrating Factor Runge–Kutta WENO Methods |
| title_short | Krylov SSP Integrating Factor Runge–Kutta WENO Methods |
| title_sort | krylov ssp integrating factor runge kutta weno methods |
| topic | strong stability preserving integrating factor Runge–Kutta weighted essentially non-oscillatory methods Krylov subspace approximation |
| url | https://www.mdpi.com/2227-7390/9/13/1483 |
| work_keys_str_mv | AT shanqinchen krylovsspintegratingfactorrungekuttawenomethods |