Hermite Upwind Positive Schemes (HUPS) for non-linear hyperbolic systems of conservation laws

Abstract

Abstract This work aims at designing a numerical model for discretizing, over compact stencils, non-linear hyperbolic systems of conservation laws, with high-order accuracy, while maintaining a positivity property of the discretization. Starting from a multi-dimensional finite volume discretization of the PDEs, we build an Hermitian polynomial based upon a least-square procedure. This polynomial is then used to interpolate, with a fourth-order accuracy, the pointwise quantities (variable and its first derivatives) that are necessary to compute the numerical fluxes at cell interfaces. A monotonicity-preserving algorithm helps to enforce the positivity of the resulting spatial discretization. Each variable of the problem and its first derivatives are then evolved in time by using a fourth-order positive Runge–Kutta algorithm (SSPRK). The resulting scheme is a fourth-order least-square Hermitian Upwind Positive Scheme (HUPS). Extensive numerical 2D tests for scalar or Euler equations of gas dynamics are driven in order to assess the potentialities of the method.