Invariant domain preserving discretization-independent schemes and convex limiting for hyperbolic systems
For computational scientists solving hyperbolic PDEs, this provides a general framework ensuring solution bounds are preserved, but the novelty is incremental as it builds on existing invariant domain and limiting concepts.
The paper introduces a discretization-independent, invariant domain preserving approximation technique for nonlinear hyperbolic systems with sources, achieving first-order accuracy, with higher-order extensions corrected via convex limiting to enforce user-specified invariant domain properties.
We introduce an approximation technique for nonlinear hyperbolic systems with sources that is invariant domain preserving. The method is discretization-independent provided elementary symmetry and skew-symmetry properties are satisfied by the scheme. The method is formally first-order accurate in space. A series of higher-order methods is also introduced. When these methods violate the invariant domain properties, they are corrected by a limiting technique that we call convex limiting. After limiting, the resulting methods satisfy all the invariant domain properties that are imposed by the user (see Theorem~7.24). A key novelty is that the bounds that are enforced on the solution at each time step are necessarily satisfied by the low-order approximation.