Invariant domain preserving discretization-independent schemes and convex limiting for hyperbolic systems

Abstract 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. Then, we introduce a series of higher-order methods. 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) and is formally high-order accurate. The two key novelties are that (i) limiting is done by enforcing bounds on quasiconcave functionals; (ii) the bounds that are enforced on the solution at each time step are necessarily satisfied by the low-order approximation.

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