Compact third-order accurate, positive preserving and divergence-free scheme

Abstract

One important problem of higher order conservative schemes is the generation of negative density and pressure which are not only non-physical, but may also blow up the numerical solution. In addition, for a multidimensional magnetohydrodynamics (MHD) solver, the solver may produce magnetic monopoles which may destabilize the solver too. To resolve the above issues, Waagan's positive pressure and density preserving scheme has been modified using the compact third-order TVD slope limiter for better accuracy and oscillation control. The resultant scheme is then integrated with the upwind constrained transport method to improve its stability and validity by keeping the solver free from magnetic monopoles. Additionally, energy correction from constrained transport method can be employed to improve the new positive scheme's positive pressure preserving properties. The new positive solver is then demonstrated on different test cases like the shock tubes, rotor problems and blast wave problems. The proposed scheme has been shown to be able to preserve positive pressure and density for flows with low density, very small plasma β and/or large pressure gradients while maintaining its high order accuracy.