Numerical boundary flux functions that give provable bounds for nonlinear initial boundary value problems with open boundaries

TL;DR

The paper develops a nonlinear entropy-based framework to design provably bounded open-boundary fluxes for nonlinear hyperbolic problems, recasting boundary penalties as numerical fluxes that bound the entropy by external data. By diagonalizing a boundary matrix and constructing nonlinear characteristic variables, the authors derive regime-specific flux functions for Burgers' equation and the two-dimensional shallow water equations, supported by a congruence transformation that preserves entropy structure. Implemented within a split-form DGSEM, these fluxes guarantee entropy stability and demonstrate superior robustness over linear-boundary approaches, including in geostrophic adjustment tests where traditional methods fail. This work offers a practical path to robust open-boundary simulations in nonlinear hyperbolic systems and points toward extensions to broader classes of equations such as the compressible Euler system.

Abstract

We present a strategy for interpreting nonlinear, characteristic-type penalty terms as numerical boundary flux functions that provide provable bounds for solutions to nonlinear hyperbolic initial boundary value problems with open boundaries. This approach is enabled by recent work that found how to express the entropy flux as a quadratic form defined by a symmetric boundary matrix. The matrix formulation provides additional information for how to systematically design characteristic-based penalty terms for the weak enforcement of boundary conditions. A special decomposition of the boundary matrix is required to define an appropriate set of characteristic-type variables. The new boundary fluxes are directly compatible with high-order accurate split form discontinuous Galerkin spectral element and similar methods and guarantee that the solution is entropy stable and bounded solely by external data. We derive inflow-outflow boundary fluxes specifically for the Burgers equation and the two-dimensional shallow water equations, which are also energy stable. Numerical experiments demonstrate that the new nonlinear fluxes do not fail in situations where standard boundary treatments based on linear analysis do.

Figures (4)

• Figure 1: The solution \ref{['eq:burgersMMS']} of Burgers' equation at final time $t=120$ with polynomials of degree seven on a five element mesh.
• Figure 2: (a): Domain for the solution \ref{['eq:mms_swe']} with a constant velocity through a channel, curved inflow-outflow boundaries and slip walls. (b): Subcritical solution \ref{['eq:mms_swe']} with $h_0 = 32$ at $t=6$. (c): Supercritical solution \ref{['eq:mms_swe']} with $h_0 = \frac{3}{5}$ at $t = 6$. The mesh contains 140 quadrilateral elements with polynomials of degree $N=5$ in each direction. All internal interfaces use the dissipation-free EC flux.
• Figure 3: Geostrophic adjustment solution at six times where the subcritical outflow boundary is imposed with the new boundary flux \ref{['eq:subcrit-outflow']} and the EC flux used at interior interfaces. The mesh is $32 \times 32$ Cartesian elements with polynomials of degree $N=8$ in each direction. (a)--(c): Initial condition and propagation of gravitational waves as they first interact with the outflow boundary where there are some artificial reflections. (d)--(f): Rotating, elliptical solution at later times.
• Figure 4: Geostrophic adjustment solution at six times where the subcritical outflow boundary is imposed from linear analysis tools that penalizes the Riemann invariants. The LLF flux is used at the physical boundaries as well as interior interfaces, the latter introduces sufficient dissipation for the simulation to successfully run to the final time. The mesh is $32 \times 32$ Cartesian elements with polynomials of degree $N=8$ in each direction. (a)--(c): Initial condition and propagation of gravitational waves as they first interact with the outflow boundary where there are significant artificial reflections. (d)--(f): Rotating, elliptical solution at later times polluted with reflections from the outflow boundaries.

Theorems & Definitions (9)

• Theorem 4.1
• proof
• Theorem 4.2
• proof
• Remark 4.3
• Remark 5.1
• Remark 5.2
• Remark 6.1
• Remark 6.2