On general and complete multidimensional Riemann solvers for nonlinear systems of hyperbolic conservation laws

TL;DR

The work addresses robust, high-order multidimensional solvers for nonlinear hyperbolic conservation laws on unstructured Voronoi-like meshes. It builds two complete solvers—the genuinely multidimensional Osher-type flux and a corner-flux upwind flux derived from the residual distribution framework—and couples them with CWENO spatial reconstruction and ADER time stepping to achieve high-order accuracy. The methods preserve important features such as stationary shear waves and demonstrate carbuncle-free behavior across diverse Euler-test problems, including 2D Riemann problems and hypersonic flows. The approaches offer a scalable, geometry-agnostic framework for multidimensional upwinding on general meshes with significant potential impact on simulation fidelity for complex flows.

Abstract

In this work, we introduce a framework to design multidimensional Riemann solvers for nonlinear systems of hyperbolic conservation laws on general unstructured polygonal Voronoi-like tessellations. In this framework we propose two simple but complete solvers. The first method is a direct extension of the Osher-Solomon Riemann solver to multiple space dimensions. Here, the multidimensional numerical dissipation is obtained by integrating the absolute value of the flux Jacobians over a dual triangular mesh around each node of the primal polygonal grid. The required nodal gradient is then evaluated on a local computational simplex involving the d+1 neighbors meeting at each corner. The second method is a genuinely multidimensional upwind flux. By introducing a fluctuation form of finite volume methods with corner fluxes, we show an equivalence with residual distribution schemes (RD). This naturally allows to construct genuinely multidimensional upwind corner fluxes starting from existing and well-known RD fluctuations. In particular, we explore the use of corner fluxes obtained from the so-called N scheme, i.e. the Roe's original optimal multidimensional upwind advection scheme. Both methods use the full eigenstructure of the underlying hyperbolic system and are therefore complete by construction. A simple higher order extension up to fourth order in space and time is then introduced at the aid of a CWENO reconstruction in space and an ADER approach in time, leading to a family of high order accurate fully-discrete one-step schemes based on genuinely multidimensional Riemann solvers. We present applications of our new numerical schemes to several test problems for the compressible Euler equations of gas-dyanamics. In particular, we show that the proposed schemes are at the same time carbuncle-free and preserve certain stationary shear waves exactly.

Figures (15)

• Figure 1: In this Figure we depict the polygonal Voronoi-like cell $\Omega_c$ and, on the right, we highlight the edge $\partial\Omega_{ac}$ between the neighboring Voronoi cells $\Omega_c$ (colored in light blue) and $\Omega_a$ (colored in blue). Note that the vertexes of the Delaunay mesh, i.e. the Voronoi generators ${a,b,c}$, are depicted with point symbols and the vertexes of the Voronoi ${p,q,r}$, i.e. the barycenters of the dual triangles, are depicted with star symbols.
• Figure 2: In this Figure we sketch the edge normal vectors $\mathbf{n}_{ac}$ and $\mathbf{n}_{bc}$ and the corner normal pointing from cell $\Omega_c$ to point $p$ which is denoted by $\mathbf{n}_{pc}$. Note that in general the intersection of edge $\overline{ac}$ with $\overline{pq}$ is not in the mid point of the edge $\overline{ac}$. Likewise for the intersection of the edge $\overline{bc}$ with $\overline{pr}$.
• Figure 3: In this Figure we show the triangle $\hat{T}_p$ used to compute the residual distribution fluctuations $\boldsymbol{\phi}_p$ around each vertex $p$ of a polygonal element $c$.
• Figure 4: Illustration of the different reconstruction stencils used for the CWENO reconstruction of order three ($M=2$) with a safety factor of $f=1.5$ for a pentagonal central element $\Omega_c$ in blue. Top-left: central stencil needed for the reconstruction of $\mathbf{P}_{\mathrm{opt}}$ (in light blue). In the other panels we report the 5 sectorial stencils containing the element itself and two consecutive neighbors needed to reconstruct piecewise linear sectorial polynomials.
• Figure 5: Steady contact wave. On the left we show the discontinuous initial density profile, which is not aligned with the mesh. Then we show the 1d cut along $y=0$ of the final density $\rho$ (middle) and the final $x-$component of the velocity$u$ (right) obtained with our novel multidimensional solvers and some classical 1d solvers. Note that the not aligned steady contact wave is maintained with machine precision.

Theorems & Definitions (2)

• Proposition 1: Equivalence of the entropy condition in potential/fluctuation form
• proof