Blended High-order Compact Difference Schemes For Solving Parabolic Problems

The parabolic problem is a type often dealt with in partial differential equations and has a very wide range of applications in the field of seepage,diffusion and heat conduction,etc.Due to the fact that the practical problems are often complicated and can not be theoretically exact solutions,it is of great theoretical significance and practical value to find numerical solutions with high precision,relatively small compution and memory.In this paper,a hybrid high-precision compact difference scheme for solving parabolic problems is established.Firstly,based on the Taylor series expansion,a four-order mixed compact difference scheme for a one-dimensional(1D)steady convection-diffusion reaction equation is constructed,and the truncation error of the difference scheme is analyzed,the results show that the theoretical precision of the scheme is fourth order.Based on the 1D stationary problem,the remainder modification method of Taylor series expansion is adopted and two kinds of unconditionally stable hybrid high-precision compact difference schemes for solving 1D parabolic problems are established.One scheme is fourth-order precision in space and time,the other one is sixth-order precision in space and third-order in time.For 2D parabolic problems,which is discrete in space using the fourth-order compact difference scheme,is discretely time-shifted by the fourth-order backward Euler's formula,resulting in a high-precision difference scheme that is fourth-order in time and space.Finally,some numerical examples with exact solutions are validated and compared with the numerical results from other schemes in the reference,verify the accuracy and reliability of the scheme in this artical.The compact difference scheme can well simulate the large Reynolds number problem,which is also one of the advantages of this schems compared to other schemes.