Third-order unconditional positivity-preserving schemes for reactive flows keeping both mass and mole balance

Abstract

In this paper, the previously proposed second-order process-based modified Patankar Runge-Kutta schemes are extended to the third order of accuracy. Owing to the process-based implicit handling of reactive source terms, the mass conservation, mole balance and energy conservation are kept simultaneously while the positivity for the density and pressure is preserved unconditionally even with stiff reaction networks. It is proved that the first-order truncation terms for the Patankar coefficients must be zero to achieve a prior third order of accuracy for most cases. A two-stage Patankar procedure for each Runge-Kutta step is designed to eliminate the first-order truncation terms, accomplish the prior third order of accuracy and maximize the Courant number which the total variational diminishing property requires. With the same approach as the second-order schemes, the third-order ones are applied to Euler equations with chemical reactive source terms. Numerical studies including both 1D and 2D ordinary and partial differential equations are conducted to affirm both the prior order of accuracy and the positivity-preserving property for the density and pressure.