Existence of discrete shock profiles of a class of monotonicity preserving schemes for conservation laws

When shock speed s times ∆t/∆x is rational, the existence of solutions of shock profile equations on bounded intervals for monotonicity preserving schemes with continuous numerical flux is proved. A sufficient condition under which the above solutions can be extended to −∞ < j < ∞, implying the existence of discrete shock profiles of numerical schemes, is provided. A class of monotonicity preserving schemes, including all monotonicity preserving schemes with C1 numerical flux functions, the second order upwinding flux based MUSCL scheme, the second order flux based MUSCL scheme with LaxFriedrichs’ splitting, and the Godunov scheme for scalar conservation laws are found to satisfy this condition. Thus, the existence of discrete shock profiles for these schemes is established when s∆t/∆x is rational.