Legendre-Petrov-Galerkin Spectral Element Method For Third And Fifth-Order Differential Equations

As one of the main methods of the numerical solution of partial differentialequation, the spectral (element) method has become attractive among scholars due toits advantages of high accuracy and geometrical flexibility.In this paper, we consider the Legendre-Petrov-Galerkin(LPG) spectral elementmethod for the third and fifth-order differential equations in a finite interval. Themain results of the paper can be summarized as follows:(1) The LPG spectral element method is proposed to solve the third-orderdifferential equations. First, we decomposed the computational domain into severalsub-domains, we then give the approximation formulation of the problem; Secondly,we consider the approximation on each sub-domain by constructing suitable interiorbasis functions. Appropriate interface basis functions are constructed to coupled allthe sub-problems. Finally, A fast resolution algorithm is proposed to solve thealgebraic system. In order to proof the convergence, we need to introduce someprojection operators were introduced and then used in the convergenc analysis.Numerical results show the high accuracy of the algorithm.(2) The LPG spectral element method is presented to approximate thethird-order partial differential equations. In the approximation, Crank-Nicolson(C-N)scheme and Crank-Nicolson leap-frog(CN-LF) scheme are use in the time advancingrespectively for Linear partial differential equation and nonlinear Korteweg-deVries(KDV) equation. As a result, we only need to solve a linear third-order ordinarydifferential equation at each time step. The ordinary differential equation is thensolved by spectral element method. Numerical results show that the method isfeasible.(3) Similar to the approximation of the third-order differential equation, weuse the LPG spectral element method to solve the fifth-order differential equation.By constructing appropriate interior and interfacical basis functions, we get a sparselinear system, which can also be solved by the fast algorithm. Numerical resultsshow that the method is also very effective to solve high odd differential equations.