High-Accuracy Compact Difference Schemes For Solving The Two-Dimensional Wave Equation

For two-dimensional wave equation,we consider the study of the difference scheme of its numerical solution.This paper mainly proposes high-order compact differential scheme by combining alternating direction implicit(ADI),explicit scheme with high-accuracy fourth-order and sixth-order compact scheme.For the two-dimensional wave equation,the calculation area is divided and the high-order compact scheme to improve the accuracy of time and space is briefly introduced,on this basis,the concrete contents are as follows:Firstly,a high-order compact difference scheme is proposed to solve two-dimensional constant coefficient wave equation.First of all,we use central difference scheme and the sixth-order compact scheme to discrete time and spatial derivatives,so an explicit scheme with the accuracy of the sixth order of space and the second order of time is obtained.In the next place,in order to achieve the high-order approximation of the time layer,the time accuracy is increased from the second order to the fourth order by the Richardson extrapolation method.The the stability of the scheme is analyzed by Fourier method.In the final of this chapter,two numerical examples are solved to demonstrate the accuracy of the new scheme.Secondly,for the above two-dimensional constant coefficient wave equation,we propose a high-order compact ADI scheme.In the first instance,the space and time derivatives are approached by the fourth-order compact scheme and central difference scheme separately.Then the scheme of the fourth-order and second-order accuracy in space and time is obtained,the Richardson extrapolation method improves accuracy to the sixth and fourth order respectively.In the next place,The Fourier method is used to verify the unconditional stability of the scheme.Afterward,the multiple-grid method is adopted,which improves the efficiency of computations.At the end of this chapter,the stability and efficiency of this scheme can be proved through analyzing numerical results.Finally,two high-order compact ADI schemes are proposed to solve two-dimensional variable coefficient wave equation.Step one,we consider the combination of compact scheme and differential scheme,which is the fourth-order of space and time accuracy.Then a compact ADI scheme with the accuracy of the fourth-order is obtained,the stability of the scheme is analyzed by energy method.Moreover,the accuracy of the space is improved to the sixth order by using the sixth-order compact scheme,which is based on the time accuracy remains unchanged.Another compact ADI scheme with the sixth-order and fourth-order accuracy in space and time is proposed separately.After that,the coefficient is adjusted correspondingly,that is,the maximum value in the calculation area is considered as the constant coefficient.Eventually,according to Fourier method,the stability analysis is achieved,the numerical accuracy of the above two ADI schemes are demonstrated by analyzing the results.