Efficient solver of relativistic hydrodynamics with implicit Runge-Kutta method
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866911837956603904 |
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| author | Touroux, Nathan Kitazawa, Masakiyo Murase, Koichi Nahrgang, Marlene |
| author_facet | Touroux, Nathan Kitazawa, Masakiyo Murase, Koichi Nahrgang, Marlene |
| contents | We propose a new method to solve the relativistic hydrodynamic equations based on implicit Runge-Kutta methods with a locally optimized fixed-point iterative solver. For numerical demonstration, we implement our idea for ideal hydrodynamics using the one-stage Gauss-Legendre method as an implicit method. The accuracy and computational cost of our new method are compared with those of explicit ones for the (1+1)-dimensional Riemann problem, as well as the (2+1)-dimensional Gubser flow and event-by-event initial conditions for heavy-ion collisions generated by TrENTo. We demonstrate that the solver converges with only one iteration in most cases, and as a result, the implicit method requires a smaller computational cost than the explicit one at the same accuracy in these cases, while it may not converge with an unrealistically large $Δt$. By showing a relationship between the one-stage Gauss-Legendre method with the iterative solver and the two-step Adams-Bashforth method, we argue that our method benefits from both the stability of the former and the efficiency of the latter. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_12696 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Efficient solver of relativistic hydrodynamics with implicit Runge-Kutta method Touroux, Nathan Kitazawa, Masakiyo Murase, Koichi Nahrgang, Marlene Nuclear Theory High Energy Physics - Phenomenology Computational Physics Fluid Dynamics We propose a new method to solve the relativistic hydrodynamic equations based on implicit Runge-Kutta methods with a locally optimized fixed-point iterative solver. For numerical demonstration, we implement our idea for ideal hydrodynamics using the one-stage Gauss-Legendre method as an implicit method. The accuracy and computational cost of our new method are compared with those of explicit ones for the (1+1)-dimensional Riemann problem, as well as the (2+1)-dimensional Gubser flow and event-by-event initial conditions for heavy-ion collisions generated by TrENTo. We demonstrate that the solver converges with only one iteration in most cases, and as a result, the implicit method requires a smaller computational cost than the explicit one at the same accuracy in these cases, while it may not converge with an unrealistically large $Δt$. By showing a relationship between the one-stage Gauss-Legendre method with the iterative solver and the two-step Adams-Bashforth method, we argue that our method benefits from both the stability of the former and the efficiency of the latter. |
| title | Efficient solver of relativistic hydrodynamics with implicit Runge-Kutta method |
| topic | Nuclear Theory High Energy Physics - Phenomenology Computational Physics Fluid Dynamics |
| url | https://arxiv.org/abs/2306.12696 |