A local pseudo arc-length method for hyperbolic conservation laws

A local pseudo arc-length method (LPALM) for solving hyperbolic conservation laws is presented in this paper. The key idea of this method comes from the original arc-length method, through which the critical points are bypassed by transforming the computational space. The method is based on local changes of physical variables to choose the discontinuous stencil and introduce the pseudo arc-length parameter, and then transform the governing equations from physical space to arc-length space. In order to solve these equations in arc-length coordinate, it is necessary to combine the velocity of mesh points in the moving mesh method, and then convert the physical variable in arclength space back to physical space. Numerical examples have proved the effectiveness and generality of the new approach for linear equation, nonlinear equation and system of equations with discontinuous initial values. Non-oscillation solution can be obtained by adjusting the parameter and the mesh refinement number for problems containing both shock and rarefaction waves.