A numerical method for solving time-dependent convection-diffusion problems

Abstract

In this paper, we develop a new numerical method for solving a timedependent convection-diffusion equation with Dirichlet's type boundary conditions. We first propose the θ-method, θ ∈ [1/2, 1] (θ = 1 corresponds to the back-ward Euler method and θ = 1/2 corresponds to the Crank-Nicolson method) to discretize the temporal variable, resulting in a linear partial differential equation (PDE). To numerically solve this linear PDE, we develop and we analyze a new cubic spline collocation method for the spatial discretization. To solve the discretized linear system, we design a collocation method and we prove that the method is second order convergent. The computed results are compared wherever possible with those already available in the literature.