A numerical method for parabolic complementarity problem

In this paper, we study the numerical solution of a parabolic complementarity problem which is a widely used model in many fields, such as option pricing, risk measures, etc. Using a power penalty method we represent the complementarity problem as a nonlinear parabolic partial differential equation (PDE). Then, we use the trapezoidal rule as the time discretization, for which we have to solve a nonlinear equation at each time step. We solve such a nonlinear equation by the fixed-point iteration and in this methodology solving a tridiagonal linear system is the major computation. We present an efficient backward substitution algorithm to handle this linear system. Numerical results are given to illustrate the advantage of the proposed algorithm (compared to the built-in command backslash in Matlab) in terms of CPU time.