High-order Compact Difference Schemes And Adaptive Method For Singular Degenerate Diffusion-reaction Equations

For the singular degenerate diffusion-reaction equations, studying blow up time and space location of their solutions has important theoretical significance and practical application value. There are many numerical methods for solving these equations. In the present research work on them, some are based on the implementation on uniform grid; while some are based on the implementation of low-precision algorithm of adaption, but the literature about gird adaptive methods based on high order compact difference schemes for solving singular degenerate diffusion-reaction equation is very few.When high order compact scheme on non-uniform grid is used to solve the problems with large gradients, its accuracy is prior to the high order compact scheme on uniform grid. In this paper, firstly, by using the truncation error remainder amendment method, we establish a high order compact difference scheme for solving the one-dimensional singular degenerate diffusion-reaction equation based on non-uniform grid. According to the derivation process, we can know that it is the second order accuracy for time, and the third order to fourth order accuracy for spare. The scheme is two-level and has three grid points in each tempral level. Thmas algorithm can be directly used to solve the equation. When time derivative about the temperature causes a significant jump at blasting point, we utilize the equal distribution principle to establish an adaptive method for both time and space. As the mesh moves, we use the line law to transfer the function values from old grids to new ones. And we extend the method for one-dimensional problems to high-dimensional problems.Finally, the stability and accuracy of the adaptive method are verified through numerical examples with exact solutions. Then the present adaptive method is used to simulate the blow up problems and to compare with the existing numerical results. It shows that the computed results of this method is close to the solutions in literature.