An operator splitting method for nonlinear convection-diffusion equations

We present a semi-discrete method for constructing approximate solutions to the initial value problem for the m-dimensional convection-diffusion equation u(t) + del.f(u) = epsilon Delta u. The method is based on the use of operator splitting to isolate the convection part and the diffusion part of the equation. In the case m > 1, dimensional splitting is used to reduce the m-dimensional convection problem to a series of one-dimensional problems. We show that the method produces a compact sequence of approximate solutions which converges to the exact solution. Finally, a fully discrete method is analyzed, and demonstrated in the case of one and two space dimensions.