KFVM-WENO: A high-order accurate kernel-based finite volume method for compressible hydrodynamics

TL;DR

KFVM-WENO delivers a fully multidimensional, kernel-based finite volume framework for hyperbolic conservation laws, notably the compressible Euler equations, by embedding WENO adaptivity directly into a kernel/ RKHS reconstruction. The method uses linearized primitive variables or characteristic variables to decouple reconstruction from flux evaluation, a KXRCF-inspired indicator to sparsely trigger nonlinear limiting, and a positivity limiter to preserve physical states, all while enabling scalable multi-GPU MPI execution. Benchmark results show high-order accuracy on smooth problems and robust, non-oscillatory behavior on shocks and instabilities across 2D and 3D tests, with grid-independence and favorable computational efficiency. The approach simplifies multidimensional reconstruction compared to dimension-by-dimension methods and is well-positioned for extensions to AMR and unstructured grids. Overall, KFVM-WENO advances high-order, collision-free, kernel-based finite-volume solvers with practical parallel performance for complex compressible flows.

Abstract

This paper presents a fully multidimensional kernel-based reconstruction scheme for finite volume methods applied to systems of hyperbolic conservation laws, with a particular emphasis on the compressible Euler equations. Non-oscillatory reconstruction is achieved through an adaptive order weighted essentially non-oscillatory (WENO-AO) method cast into a form suited to multidimensional reconstruction. A kernel-based approach inspired by radial basis functions (RBF) and Gaussian process (GP) modeling, which we call KFVM-WENO, is presented here. This approach allows the creation of a scheme of arbitrary order of accuracy with simply defined multidimensional stencils and substencils. Furthermore, the fully multidimensional nature of the reconstruction allows for a more straightforward extension to higher spatial dimensions and removes the need for complicated boundary conditions on intermediate quantities in modified dimension-by-dimension methods. In addition, a new simple-yet-effective set of reconstruction variables is introduced, which could be useful in existing schemes with little modification. The proposed scheme is applied to a suite of stringent and informative benchmark problems to demonstrate its efficacy and utility. A highly parallel multi-GPU implementation using Kokkos and the message passing interface (MPI) is also provided.

Figures (10)

• Figure 1: The full radius-2 stencil $S_0$ is shown on the left with its five substencils, $S_q, q=1, \dots, 5$, on the right. The cell with the diamond in each (sub) stencil indicates the central cell containing $\bm{x}_*$ where reconstruction is performed. The full stencil $S_0$ has 13 cells, while each of the substencils has five.
• Figure 2: The full radius-3 stencil $\mathcal{S}_0$ is shown on the left with its five substencils, $\mathcal{S}_q, q=1, \dots, 5$, on the right. The cell with the diamond in each (sub)stencil indicates the central cell containing $\bm{x}_*$ where reconstruction is being performed. The full stencil $\mathcal{S}_0$ has 29 cells, the central substencil $\mathcal{S}_1$ has 13 cells, and each of the remaining biased substencils, $\mathcal{S}_2, \dots, \mathcal{S}_5$, has 10.
• Figure 3: Shown is a trace of the density in the Sod shock tube problem as obtained from four different cases. The solid black line shows the exact analytical solution. The dashed lines with circle and square markers show the results of the radius $R=2$ and $R=3$ schemes in the grid-aligned configuration. The dotted lines with cross and triangle markers show the results of the radius $R=2$ and $R=3$ schemes in the grid-tilted configuration. The inset shows a zoom-in of the region near contact discontinuity.
• Figure 4: Shown is the density field for the Richtmeyer-Meshkov instability at the final time of $t=3.33$ as solved by the radius $R=2$ scheme (top) and the radius $R=3$ scheme (bottom) on a grid with spacing $\Delta=1/512$. We display the view zoomed into the region $[5/2,11/2]\times [0,1]$ to highlight the interface.
• Figure 5: Shown in black are the cells flagged for WENO reconstruction at the final time for the radius $R=2$ scheme. The view has been zoomed in to match \ref{['fig:rmiR2R3']}.