High-Order Conservative Finite Difference Schemes For Several Wave Propagation Equations

In practical applications,the mathematical models that describe,explain or predict the natural phenomena and the basic laws of science are usually large-scale nonlinear coupled differential systems.However,most differential equations cannot be expressed analytically.Thus the numerical methods of nonlinear differential equations are well developed.In this paper,several classical wave propagation equations,such as Equal-Width wave equation,regularized long-wave equation(RLW)and RLW-KdV equation,are selected to construct highorder conservation numerical schemes by finite difference method.Firstly,a three-time level linear high-order conservative standard difference scheme and a three-time level linear high-order conservative compact difference scheme are constructed respectively for the Equal-Width wave equation.The conservation of energy and mass,the existence,uniqueness and boundedness of numerical solutions and the convergence and stability of the two finite difference schemes are proved by using the discrete energy method.The numerical results verify the convergence order with O(τ2+h4)in the discrete L∞-norm,conservation and stability of the two finite difference schemes.Secondly,a three-time level linear conservative compact difference scheme and a threepoint linear conservative compact difference scheme based on the reduced order method are constructed respectively for the regular long-wave equation.The conservation of the difference scheme,the unique solvability and boundedness of the solution,the convergence and stability of the difference scheme are proved by using the discrete energy method.Numerical examples verify the convergence order with O(τ2+h4)in the discrete L∞-norm of the three-time level linear conservative compact difference scheme,and the convergence order with O(τ2+h4)in the discrete L∞-norm,L2-norm and H1-norm of the linear compact difference scheme based on the reduced order method.The stability and conservation of the two difference schemes are verified.Multiple isolated wavelet collisions and wave evolution with time under Maxwellian initial condition are simulated.Finally,a high-order compact difference scheme based on the reduced order method for the RLW-KdV equation is proposed.The conservation of energy and mass,the boundedness and uniqueness of numerical solutions,the convergence and stability of the difference scheme are proved by the discrete energy method.The numerical results verify the convergence order,stability and conservation of the difference scheme with O(τ2+h4)in the discrete L∞-norm,which indicates the validity and reliability of the proposed difference scheme.