We present a general framework to design Godunov-type schemes for multidimensional ideal magnetohydrodynamic (MHD) systems, having the divergence-free relation and the related properties of the magnetic field B as built-in conditions. Our approach mostly relies on the Constrained Transport (CT) discretization technique for the magnetic field components, originally developed for the linear induc...

This study investigates the viscous term formulation for the newly developed energy stable flux reconstruction (ESFR) scheme, as an extension to the linear advection flux formulation in the original ESFR scheme. The concept of energy stability for the flux reconstruction approach is put forward. This paper also discusses the formulation of inviscid first derivative flux based on flux reconstruc...

We present well-balanced high-resolution finite volume schemes for simulating waves in the outer solar atmosphere. The schemes approximate the stratified MHD equations with an upwind discretization of the Godunov-Powell source term and a locally hydrostatic pressure reconstruction that preserves discrete steady states. The paper summarizes recent articles [4, 5, 6].

Several models in mathematical physics are described by quasilin ear hyperbolic systems with source term which in several cases may become sti Here a suitable central numerical scheme for such problems is developed and application to shallow water equations Broadwell model and Extended Thermodynamics are mentioned The numerical methods are a generalization of the Nessyahu Tadmor scheme to the n...

This paper is concerned with the generalized Allen-Cahn equation with a nonlinear mobility that can degenerate, which also includes an advection term as found in phase-field models. A class of maximum principle preserving schemes will be studied for the generalized Allen-Cahn equation, with either the commonly used polynomial free energy or the logarithmic free energy, and with a nonlinear dege...