Fast Higher-Order Interpolation and Restriction in ExaHyPE Avoiding Non-physical Reflections

TL;DR

This work enhances AMR-based wave simulations by replacing the baseline trilinear interpolation with higher-order, matrix-based interpolation and restriction constructed from Taylor expansions around coarse-cell centres. By collapsing multiple face configurations into a compact reference-mconfiguration and using explicit rotations, the method achieves improved accuracy and removes AMR-boundary errors that previously caused Hamiltonian constraint violations in stationary black-hole tests. The approach delivers convergence consistent with the chosen order (second and third order) and demonstrates stability gains in long-term evolutions, at the cost of larger matrix construction costs and CSR-format limitations. The results suggest substantial practical impact for high-fidelity, adaptively refined relativity simulations, with clear avenues for optimization and GPU porting in future work.

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.

Figures (5)

• Figure 1: A diagram of the coarse and fine face structures at an AMR boundary ExaGRyPE
• Figure 2: The error convergence plot for each interpolation scheme. The logarithm of the error norm, $\log(\epsilon)$, is plotted against $log(h)$. The order of convergence is obtained by comparing this to plots of $h^p$ where $p$ is a possible order of convergence.
• Figure 3: Plots of the Hamiltonian constraint errors along the domain's x axis for a simulation of a stationary black hole using the tensor product and second order interpolations and restriction.
• Figure 4: Left: Runtime for each interpolation method against patch size. Right: Time spent in matrix construction for higher order interpolation compared to total run-time
• Figure 5: Three possibles arrangements of the coarse cells being considered for the second order interpolation. The shaded ellipse indicates the coarse cell containing the current fine cells being interpolated. Any dotted lines indicate the edge of a face. Through reflections and axis permutation, these three arrangements can be modified to encompass any possible scenario. Left: The default case in the centre of the face. Centre: The case where the centre coarse cell is against one face boundary. Right: The case where the centre coarse cell is against two face boundaries.