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

When shock speed $s$ times $\Delta t/\Delta x$ 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 $-\infty <j<\infty$, implying the existence of discrete shock profiles of numerical schemes, is provided. A class of monotonicity preserving schemes, including all monotonicity preserving schemes with $C^{1}$ numerical flux functions, the second order upwinding flux based MUSCL scheme, the second order flux based MUSCL scheme with Lax-Friedrichs' 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 $s\Delta t/\Delta x$ is rational.