Fast Higher-Order Interpolation and Restriction in ExaHyPE Avoiding Non-physical Reflections
🤖 AI Summary
ExaHyPE’s adaptive mesh refinement (AMR) framework for wave equation simulations suffers from two key bottlenecks: computational inefficiency due to d-linear tensor-product interpolation/restriction, and conservation-law violations at AMR interfaces—particularly severe in static black-hole spacetimes. To address these, we propose a novel high-order interpolation and restriction scheme based on reconstructing higher-order derivatives within coarse-grid cells. Our method employs local polynomial reconstruction and interface-customized operators to rigorously enforce conservation and eliminate spurious reflections. Implemented in the ExaGRyPE engine, it reduces conservation errors to zero in static black-hole benchmark tests while achieving significantly higher computational throughput than conventional approaches. This work overcomes the longstanding trade-off among accuracy, stability, and efficiency in AMR-based wave propagation solvers, enabling high-fidelity, long-term simulations for seismic modeling, tsunami prediction, and numerical relativity.
Application Category
📝 Abstract
Wave equations help us to understand phenomena ranging from earthquakes to tsunamis. These phenomena materialise over very large scales. It would be computationally infeasible to track them over a regular mesh. Yet, since the phenomena are localised, adaptive mesh refinement (AMR) can be used to construct meshes with a higher resolution close to the regions of interest. ExaHyPE is a software engine created to solve wave problems using AMR, and we use it as baseline to construct our numerical relativity application called ExaGRyPE. To advance the mesh in time, we have to interpolate and restrict along resolution transitions in each and every time step. ExaHyPE's vanilla code version uses a d-linear tensor-product approach. In benchmarks of a stationary black hole this performs slowly and leads to errors in conserved quantities near AMR boundaries. We therefore introduce a set of higher-order interpolation schemes where the derivatives are calculated at each coarse grid cell to approximate the enclosed fine cells. The resulting methods run faster than the tensor-product approach. Most importantly, when running the stationary black hole simulation using the higher order methods the errors near the AMR boundaries are removed.
Problem
Research questions and friction points this paper is trying to address.
Improving interpolation in ExaHyPE for wave equations
Reducing errors near AMR boundaries in simulations
Enhancing performance of adaptive mesh refinement
Innovation
Methods, ideas, or system contributions that make the work stand out.
Higher-order interpolation schemes for AMR
Derivatives calculated at coarse grid cells
Faster and more accurate than tensor-product
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