High-efficiency implicit scheme for solving first-order partial differential equations

Abstract

We present three new approaches for solving first-order quasi-linear partial differential equations (PDEs) with iterative methods of high stability and low cost. The first is a new numerical version of the method of characteristics that converges efficiently, under certain conditions. The next two approaches initially apply the unconditionally stable Crank–Nicolson method, which induces a system of nonlinear equations. In one of them, we solve this system by using the first optimal schemes for systems of order four (Ermakov’s Hyperfamily). In the other approach, using a new technique called JARM decoupling, we perform a modification that significantly reduces the complexity of the scheme, which we solve with scalar versions of the aforementioned iterative methods. This is a substantial improvement over the conventional way of solving the system. The high numerical performance of the three approaches is checked when analyzing the resolution of some examples of nonlinear PDEs.