High-order Compact Finite Difference Method For The Incompressible Magnetohydrodynamics Equations

Studies on numerical methods for solving the incompressible magnetohydrodynamics(MHD)equations attract much concerns of a great many researchers due to theirs' important theoretical value and practical significance.This thesis focuses on the numerical solution of the MHD problems by finite difference method.Fourth-and sixth-order accurate compact finite difference schemes are developed,respectively,and the advantages of high accuracy,good stability and high efficiency are validated by some numerical experiments.Firstly,start from the MHD equations in form of the original variables,the steady and unsteady stream-vortex-current density equations of them are derived.For the steady case,the space term is expanded by using the Taylor series,combined with the ideas of explicit compact difference and implicit compact difference schemes,the fourth-order Pade formulas are used for the discretization of the first derivatives,a high-order compact difference scheme is proposed for solving the two-dimensional steady MHD equations with stream-vortex-current density formulation.It is marked as HOC4 in the paper and the truncation error is analyzed.Afterwards,based on the construction of the HOC4 scheme,the fifth-order and sixth-order derivatives in the truncation error terms are replaced by the unknown function and the linear combination of the first-order and second-order derivatives,and the values of the first-order and second-order derivatives are computed by the combined compact difference(CCD)schemes,a sixth-order accurate compact difference(HOC6)scheme for solving the two-dimensional steady MHD equations with stream-vortex-current density formulation is derived.Analysis of the truncation error is also conducted.Secondly,for the two-dimensional unsteady MHD equations with stream-vortex-current density formulation,the space term is expanded by using the Taylor series,and the first derivative is used as the unknown quantity and computed by the fourth-order Pade difference scheme,the time derivative term is discretized by the unconditionally stable second-order backward difference formula,and the boundaries are also discretized by the fourth-order difference schemes,a temporally second-order and spatially fourth-order accurate compact difference scheme is proposed.The scheme is marked as HOC(2,4)in the paper and then is used to solve the MHD problems with analytical solutions and to numerically simulate the magnetically driven square cavity flow problems.Afterwards,to match time accuracy with that of space,the unconditionally stable fourth-order backward difference formula is used to improve temporal accuracy to the fourth-order and new scheme is denoted as HOC(4,4).The MHD problems with analytical solutions and the magnetically driven square cavity flow problems are also solved and simulated to demonstrate the accuracy,stability and effectiveness of the present method.Thirdly,still focus on the two-dimensional unsteady MHD equations with stream-vortex-current density formulation,the space term is expanded by Taylor series,combined with the idea of explicit compact difference and implicit compact difference schemes,the first-order and the second derivatives are used as unknowns together with the unknown function to construct a sixth-order compact difference scheme,in which,the first and second derivatives are computed by the CCD schemes.The time derivative term is discretized by the unconditionally stable third-order backward difference formula.And the boundaries are calculated by the sixth-order difference formula.The scheme is marked as HOC(3,6),which means it is the third-order in time and sixth-order in space.Numerical experiments of the MHD equations with exact solutions and magnetically driven square cavity flow problems are conducted to illustrate its performances.In order to match the accuracy of time with space,the unconditionally stable sixth-order backward difference scheme is used for discretization of the time term and.a temporally and spatially sixth-order accurate compact difference scheme,which is marked as HOC(6,6)scheme,is proposed.At the same time,some numerical examples about the MHD problems with analytical solutions are computed to show the accuracy and stability of the present scheme and the magnetically driven square cavity flow problems are numerically simulated.Finally,summaries and conclusions about this thesis are drawn and some ideas and plans for further researches are also discussed.