High-order positivity-preserving well-balanced schemes: combining flux globalization with modified Patankar deferred correction methods

Positivity-preserving schemes for hyperbolic balance laws are typically built from the Zhang-Shu limiter and strong-stability-preserving Runge-Kutta (SSP-RK) time stepping, at the cost of a restrictive time-step condition. The modified Patankar deferred correction (mPDeC) method removes this restriction, offering high-order accuracy, conservation, and unconditional positivity. However, its solution-dependent Patankar weights make the time update nonlinear, so, unlike with RK methods, preserving (steady and moving) equilibria, i.e., well-balancing (WB), is no longer straightforward. In this paper, we construct high-order, WB, positivity-preserving mPDeC schemes by combining mPDeC with a flux-globalization approach. We demonstrate the idea on the one-dimensional shallow water equations, using a WB A-WENO spatial discretization, and introduce a modified local characteristic decomposition that remains stable in the presence of dry states. Numerical experiments on one-dimensional benchmarks confirm that the method satisfies the desired properties.

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