Positivity-preserving high-resolution schemes for systems of conservation laws

A new class of flux-limited schemes for systems of conservation laws is presented that is both high-resolution and positivity-preserving. The schemes are obtained by extending the Steger–Warming method to second-order accuracy through the use of component-wise TVD flux limiters while ensuring that the coefficients of the discretization equation are positive. A coefficient is considered positive if it has all-positive eigenvalues and has the same eigenvectors as those of the convective flux Jacobian evaluated at the corresponding node. For certain systems of conservation laws, such as the Euler equations for instance, this condition is sufficient to guarantee positivity-preservation. The method proposed is advantaged over previous positivity-preserving flux-limited schemes by being capable to capture with high resolution all wave types (including contact discontinuities, shocks, and expansion fans). Several test cases are considered in which the Euler equations in generalized curvilinear coordinates are solved in 1D, 2D, and 3D. The test cases confirm that the proposed schemes are positivity-preserving while not being significantly more dissipative than the conventional TVD methods. The schemes are written in general matrix form and can be used to solve other systems of conservation laws, as long as they are homogeneous of degree one.