Local pseudo arc-length method for hyperbolic partial differential equation

In this paper, a local pseudo arc-length method is proposed for hyperbolic partial differential equation with singular problem of shock waves, and the forms of space transformation and adaptive mesh refinement are analyzed for the global pseudo arc-length method. In order to improve the computational efficiency, the local pseudo arc -length method which gives the ways to determine the position of singular points and select the computational stencil is presented according to the properties of shock wave. The modifications of the new method involve how to introduce the arc-length parameters and how to dispose the shock wave oscillation. The feasibility of the local pseudo arc-length method in capturing and tracking shock is proved through numerical examples, and the superiority of local pseudo arc-length method in dealing with hyperbolic partial differential equation is shown by comparing our method with Godunov method for disposing different initial conditions of the hyperbolic problems. The numerical results demonstrate that our new method can be applied to engineering problems.