Minimally Implicit Runge-Kutta (MIRK) Fully Well-Balanced Method for the 1D Shallow Water Equations with Topography and Manning Friction
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The Minimally Implicit Runge–Kutta (MIRK) methods are proposed as an efficient time-integration strategy for nonlinear hyperbolic balance laws with stiff source terms. Although Implicit–Explicit (IMEX) methods are widely used for such problems, their computational efficiency could be limited due to the increasing complexity of the implicit stages that must be solved. MIRK schemes overcome this limitation by retaining the stability advantages of semi-implicit approaches while drastically reducing implicit coupling, resulting in a substantially lower computational cost without sacrificing accuracy or robustness. The MIRK framework has previously been successfully applied in challenging astrophysical contexts. This contribution focuses on its application to the one-dimensional shallow water system with topography and Manning friction, where first- and second-order schemes have been developed. These methods are fully well-balanced, in the sense that they preserve stationary solutions of the system of equations, while maintaining accuracy and stability in stiff regimes. Extensive numerical experiments demonstrate that MIRK methods provide significant speed-ups compared with classical IMEX schemes, without compromising accuracy or reliability.
