ADER-WENO Finite Volume Schemes with Space-Time Adaptive Mesh Refinement (1212.3585v3)

Abstract: We present the first high order one-step ADER-WENO finite volume scheme with Adaptive Mesh Refinement (AMR) in multiple space dimensions. High order spatial accuracy is obtained through a WENO reconstruction, while a high order one-step time discretization is achieved using a local space-time discontinuous Galerkin predictor method. Due to the one-step nature of the underlying scheme, the resulting algorithm is particularly well suited for an AMR strategy on space-time adaptive meshes, i.e.with time-accurate local time stepping. The AMR property has been implemented 'cell-by-cell', with a standard tree-type algorithm, while the scheme has been parallelized via the Message Passing Interface (MPI) paradigm. The new scheme has been tested over a wide range of examples for nonlinear systems of hyperbolic conservation laws, including the classical Euler equations of compressible gas dynamics and the equations of magnetohydrodynamics (MHD). High order in space and time have been confirmed via a numerical convergence study and a detailed analysis of the computational speed-up with respect to highly refined uniform meshes is also presented. We also show test problems where the presented high order AMR scheme behaves clearly better than traditional second order AMR methods. The proposed scheme that combines for the first time high order ADER methods with space--time adaptive grids in two and three space dimensions is likely to become a useful tool in several fields of computational physics, applied mathematics and mechanics.

• The paper introduces a high-order ADER-WENO finite volume scheme incorporating space-time adaptive mesh refinement (AMR).
• It achieves high accuracy using WENO spatial reconstruction and a local space-time discontinuous Galerkin predictor, enhanced by cell-by-cell AMR with local time stepping.
• Numerical tests on Euler and MHD equations demonstrate superior performance over lower-order methods and validate the scheme's high-order accuracy for complex physical systems.

Insightful Overview of ADER-WENO Finite Volume Schemes with Space-Time AMR

The academic paper presents a significant advancement in numerical schemes for solving nonlinear systems of hyperbolic conservation laws. The authors introduce a high-order one-step ADER-WENO finite volume methodology, integrated with space-time adaptive mesh refinement (AMR). This innovative approach seeks to enhance spatial and temporal accuracy in computational physics, emphasizing applications in compressible gas dynamics and magnetohydrodynamics (MHD).

Key Features of the Scheme

The paper details several core components that constitute the new scheme:

• High-Order Spatial and Temporal Accuracy: The approach employs Weighted Essentially Non-Oscillatory (WENO) reconstruction for spatial accuracy, combined with a local space-time discontinuous Galerkin predictor for time discretization. This allows the scheme to achieve high-order accuracy without relying on multi-stage Runge-Kutta integrators.
• Adaptive Mesh Refinement (AMR): Implementing AMR 'cell-by-cell' with a standard tree-type algorithm facilitates dynamic grid adaptation while maintaining robustness through local time stepping. This is particularly useful for simulations of complex physical systems requiring localized high-resolution meshes.
• Parallelization via MPI: The scheme leverages Message Passing Interface (MPI) for efficient computation, enabling it to handle substantial computational workloads typical in high-resolution simulations.

Numerical Results and Comparisons

The proposed ADER-WENO scheme was tested across various nonlinear systems, including the Euler equations and MHD equations, demonstrating superior performance over traditional second-order methods. Numerical convergence studies confirm its high-order accuracy, providing compelling validation of its efficacy.

Implications and Future Developments

Practically, this scheme is poised to significantly impact areas such as computational fluid dynamics, astrophysics, and mechanics where resolving complex dynamical phenomena is critical. Theoretically, this work opens pathways for leveraging one-step schemes in high complexity models beyond hyperbolic conservation laws.

Potential future research could explore further extensions to turbulent viscous flows, multiphase chemically reacting flows, and nonconservative hyperbolic systems. The implementation of dynamic load balancing in MPI settings may also enhance computational efficiency in large-scale simulations.

In conclusion, this paper contributes markedly to the sophistication of numerical methods in computational physics, aiming for precision and scalability in simulations of intricate systems. Researchers stand to benefit considerably from the insights and methodologies presented herein, paving the way for more refined and reliable computational tools in scientific computing.