Adaptive mesh refinement for coupled elliptic-hyperbolic systems. (English) Zbl 1104.65092
Summary: We present a modification to the adaptive mesh refinement algorithm of M. J. Berger and J. Oliger [ibid. 53, 484–512 (1984; Zbl 0536.65071)] designed to solve systems of coupled, nonlinear, hyperbolic and elliptic partial differential equations. Such systems typically arise during constrained evolution of the field equations of general relativity. The novel aspect of this algorithm is a technique of “extrapolation and delayed solution” used to deal with the non-local nature of the solution of the elliptic equations, driven by dynamical sources, within the usual Berger and Oliger time-stepping framework.
We show empirical results demonstrating the effectiveness of this technique in axisymmetric gravitational collapse simulations, and further demonstrate that the solution time scales approximately linearly with problem size. We also describe several other details of the code, including truncation error estimation using a self-shadow hierarchy, and the refinement-boundary interpolation operators that are used to help suppress spurious high-frequency solution components (“noise”).
We show empirical results demonstrating the effectiveness of this technique in axisymmetric gravitational collapse simulations, and further demonstrate that the solution time scales approximately linearly with problem size. We also describe several other details of the code, including truncation error estimation using a self-shadow hierarchy, and the refinement-boundary interpolation operators that are used to help suppress spurious high-frequency solution components (“noise”).
MSC:
| 65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 83-08 | Computational methods for problems pertaining to relativity and gravitational theory |
| 35M20 | PDE of composite type (MSC2000) |
Keywords:
adaptive mesh refinement; coupled elliptic-hyperbolic systems; finite different methods; numerical relativity; numerical examples; error estimationCitations:
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