Finite volume method and asymptotic method for parabolic PDE involving some small parameter

Parabolic partial differential equation with small parameter in x direction and different small parameter in t direction is considered. This kind of problem leads to boundary layer phenomena in both side of x direction and bottom side of t direction. The analytical solution changes rapidly near three boundary layer. Firstly, the asymptotic solution with order one for all small parameter is studied. The analytical solution is approximated by the degenerate solution and three boundary layer functions. Secondly, non-equidistant mesh partitions both in x direction and t direction are constructed according to Shishkin´s idea. The domain is divided into the outside boundary layer and the boundary layer. Thirdly, traditional finite volume method is applied by integrating the equation over the control volume. The discretized equation is derived. Asymptotic solution is used for the outside boundary layer, while finite volume method is applied for three boundary layers. Finally, numerical example is studied to demonstrate the accuracy of the present method.