On Adaptive Time Step Selection Procedure for Parabolic Equations with the LIM Scheme

The paper deals with the construction of the time step selection procedure for the Local Iteration Modified (LIM) scheme for solving parabolic equations. This scheme has been designed for unsteady applications and presents an explicit iterative solver which is based on Chebyshev polynomials of a special construction. We propose to apply the LIM scheme to solve a parabolic equation with the known procedure of the time step selection. In this procedure the implicit scheme is used to derive the problem solution. This scheme can be efficiently implemented in a one-dimensional case but it is difficult to implement it in a general multidimensional framework. On the contrary, the LIM scheme is universal in terms of applicability to multidimensional discretizations on arbitrary grids, including unstructured grids, which is relevant in a wide class of practical applications. The proposed algorithm is as follows. The predicted solution is found by the convenient explicit scheme. A priori choice of a time step is made by estimating the approximation error of the implicit scheme: both the current LIM solution and the predicted solution are substituted into the implicit scheme formula as the solution at the lower and upper levels, respectively. With the chosen time step the solution at a new time level is computed by the LIM scheme, i.e., without application of an implicit solver. The numerical results for the one-dimensional heat conduction equation demonstrate efficiency of the LIM scheme in combination with the time step selection procedure. Generalization of the approach to the multidimensional framework is straightforward.