A spectral element method using the modal basis and its application in solving second‐order nonlinear partial differential equations

Abstract

We present a high-order spectral element method (SEM) using modal (or hierarchical) basis for modeling of some nonlinear second-order partial differential equations in two-dimensional spatial space. The discretization is based on the conforming spectral element technique in space and the semi-implicit or the explicit finite difference formula in time. Unlike the nodal SEM, which is based on the Lagrange polynomials associated with the Gauss-Lobatto-Legendre or Chebyshev quadrature nodes, the Lobatto polynomials are used in this paper as modal basis. Using modal bases due to their orthogonal properties enables us to exactly obtain the elemental matrices provided that the element-wise mapping has the constant Jacobian. The difficulty of implementation of modal approximations for nonlinear problems is treated in this paper by expanding the nonlinear terms in the weak form of differential equations in terms of the Lobatto polynomials on each element using the fast Fourier transform (FFT). Utilization of the Fourier interpolation on equidistant points in the FFT algorithm and the enough polynomial order of approximation of the nonlinear terms can lead to minimize the aliasing error. Also, this approach leads to finding numerical solution of a nonlinear differential equation through solving a system of linear algebraic equations. Numerical results for some famous nonlinear equations illustrate efficiency, stability and convergence properties of the approximation scheme, which is exponential in space and up to third-order in time. 478F. FAKHAR-IZADI AND M. DEHGHAN the finite element discretization and shares specific features with both the h-version and p-version of the finite element method, made a revolution in the spectral methods and extended their applications dramatically.Spectral element methods are essentially higher order finite element methods that exhibit spectral convergence for the smooth functions [10]. In the spectral element methods, similar to the finite element methods, a weak form of the boundary value problem is obtained, and the domain is decomposed into some elements, but the approximation of the field variables within each element is spectral. Combination of the spectral element method with a time-stepping method is often called spectral element method of lines and can be used for handling the evolutionary problems [11]. However, some authors have used the space-time spectral element method in which both space and time variables are approximated by SEM [12,13].Recently, the spectral element methods have gained more attention by researchers and have been applied in various fields of science and engineering [2,5,[14][15][16][17][18][19][20]. Authors of [18] have presented a spectral element method to price European options with jump diffusion, which employs piecewise high-order Legendre polynomial expansions to approximate the option price represented pointwise on a Gauss-Lobatto mesh within each element. The spectral element method has been developed for the solution of the Black-Scholes equation t...

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