A numerical study of Diagonally split Runge-Kutta methods for PDEs with discontinuities
Diagonally split Runge-Kutta (DSRK) time discretization methods are a class of implicit time-stepping schemes which offer both high-order convergence and a form of nonlinear stability known as unconditional contractivity. This combination is not possible within the classes of Runge-Kutta or linear m...
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| Format: | Journal article |
| Language: | English |
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2008
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| _version_ | 1797095941097914368 |
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| author | Macdonald, C Gottlieb, S Ruuth, S |
| author_facet | Macdonald, C Gottlieb, S Ruuth, S |
| author_sort | Macdonald, C |
| collection | OXFORD |
| description | Diagonally split Runge-Kutta (DSRK) time discretization methods are a class of implicit time-stepping schemes which offer both high-order convergence and a form of nonlinear stability known as unconditional contractivity. This combination is not possible within the classes of Runge-Kutta or linear multistep methods and therefore appears promising for the strong stability preserving (SSP) time-stepping community which is generally concerned with computing oscillation-free numerical solutions of PDEs. Using a variety of numerical test problems, we show that although second- and third-order unconditionally contractive DSRK methods do preserve the strong stability property for all time step-sizes, they suffer from order reduction at large step-sizes. Indeed, for time-steps larger than those typically chosen for explicit methods, these DSRK methods behave like first-order implicit methods. This is unfortunate, because it is precisely to allow a large time-step that we choose to use implicit methods. These results suggest that unconditionally contractive DSRK methods are limited in usefulness as they are unable to compete with either the first-order backward Euler method for large step-sizes or with Crank-Nicolson or high-order explicit SSP Runge-Kutta methods for smaller step-sizes. We also present stage order conditions for DSRK methods and show that the observed order reduction is associated with the necessarily low stage order of the unconditionally contractive DSRK methods. © 2007 Springer Science+Business Media, LLC. |
| first_indexed | 2024-03-07T04:35:09Z |
| format | Journal article |
| id | oxford-uuid:cfaf3047-d96b-45e9-9387-361d9077a351 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T04:35:09Z |
| publishDate | 2008 |
| record_format | dspace |
| spelling | oxford-uuid:cfaf3047-d96b-45e9-9387-361d9077a3512022-03-27T07:44:21ZA numerical study of Diagonally split Runge-Kutta methods for PDEs with discontinuitiesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:cfaf3047-d96b-45e9-9387-361d9077a351EnglishSymplectic Elements at Oxford2008Macdonald, CGottlieb, SRuuth, SDiagonally split Runge-Kutta (DSRK) time discretization methods are a class of implicit time-stepping schemes which offer both high-order convergence and a form of nonlinear stability known as unconditional contractivity. This combination is not possible within the classes of Runge-Kutta or linear multistep methods and therefore appears promising for the strong stability preserving (SSP) time-stepping community which is generally concerned with computing oscillation-free numerical solutions of PDEs. Using a variety of numerical test problems, we show that although second- and third-order unconditionally contractive DSRK methods do preserve the strong stability property for all time step-sizes, they suffer from order reduction at large step-sizes. Indeed, for time-steps larger than those typically chosen for explicit methods, these DSRK methods behave like first-order implicit methods. This is unfortunate, because it is precisely to allow a large time-step that we choose to use implicit methods. These results suggest that unconditionally contractive DSRK methods are limited in usefulness as they are unable to compete with either the first-order backward Euler method for large step-sizes or with Crank-Nicolson or high-order explicit SSP Runge-Kutta methods for smaller step-sizes. We also present stage order conditions for DSRK methods and show that the observed order reduction is associated with the necessarily low stage order of the unconditionally contractive DSRK methods. © 2007 Springer Science+Business Media, LLC. |
| spellingShingle | Macdonald, C Gottlieb, S Ruuth, S A numerical study of Diagonally split Runge-Kutta methods for PDEs with discontinuities |
| title | A numerical study of Diagonally split Runge-Kutta methods for PDEs with discontinuities |
| title_full | A numerical study of Diagonally split Runge-Kutta methods for PDEs with discontinuities |
| title_fullStr | A numerical study of Diagonally split Runge-Kutta methods for PDEs with discontinuities |
| title_full_unstemmed | A numerical study of Diagonally split Runge-Kutta methods for PDEs with discontinuities |
| title_short | A numerical study of Diagonally split Runge-Kutta methods for PDEs with discontinuities |
| title_sort | numerical study of diagonally split runge kutta methods for pdes with discontinuities |
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