High-order Compact Difference Scheme For Solving Linear Hyperbolic Equations

Hyperbolic equation is a kind of important partial differential equation.Because it is difficult to find the exact solution of the problem itself,it has far-reaching significance and practical application value to solve this kind of equation by numerical method.In this paper,a high-order compact difference scheme for solving linear hyperbolic equation is established.Firstly,Kreiss fourth-order compact difference formula is used to approximate it in space and time.Based on Taylor series expansion and truncation error correction,a high accuracy compact fully implicit scheme for solving one-dimensional linear hyperbolic equations is proposed.The scheme has fourth-order accuracy in both time and space.The stability of the scheme is analyzed by Fourier method.Several numerical examples with exact solutions are used to validate the proposed scheme.The numerical experiments show that the proposed scheme are effective in solving one-dimensional linear hyperbolic equations.Comparing with numerical methods in the literature,the present method have better stability and accuracy.Next,the high-precision compact difference method for one-dimensional linear hyperbolic equation is extended directly to two-dimensional problems,and a fourth-order compact difference scheme is established.At this time,iterative computation is needed,and a modified multi-grid full approximation scheme is adopted,which speeds up the convergence speed of iteration,reduces the iteration times,saves the computation time and improves the calculation efficiency.Some numerical examples with exact solutions are used to validated the present scheme.The numerical results show that the method in this paper can achieve fourth-order accuracy in both time and space,which is consistent with the theoretical analysis in this paper.Moreover,the calculation error is obviously smaller than that in the literature,and the calculation accuracy is higher.Finally,the scheme deduced in this paper is integrated into the software of the finite difference method for solving partial differential equations,which makes the numerical solution of partial differential equations more convenient for researchers to use the present schemes and do comparison research with the methods in this paper.