A reduced‐order extrapolated Crank–Nicolson collocation spectral method based on proper orthogonal decomposition for the two‐dimensional viscoelastic wave equations

Abstract

In order to reduce the order of the coefficient vectors of the solutions for the classical Crank–Nicolson collocation spectral (CNCS) method of two‐dimensional (2D) viscoelastic wave equations via proper orthogonal decomposition, we first establish a reduced‐order extrapolated CNCS (ROECNCS) method of the 2D viscoelastic wave equations so that the ROECNCS method has the same basis functions as the classical CNCS method and maintains all advantages of the classical CNCS method. Then, by means of matrix analysis, we discuss the existence, stability, and convergence for the ROECNCS solutions so that the theory analysis becomes very concise. Finally, we utilize some numerical experiments to validate that the consequences of numerical computations are accordant with the theoretical analysis such that the effectiveness and viability of the ROECNCS method are further verified. Therefore, both theory and method of this paper are new and totally different from the existing reduced‐order methods.

The Crank–Nicolson finite element method for the 2D uniform transmission line equation

“…Using the property of R h , (8), (19), Taylor's formula, and the Hölder and Cauchy inequalities, we get…”

“…Provided that it is applied for settling big data processing in artificial intelligence and/or computational linguistics, there will be more than tens of millions unknown numbers. Fortunately, a proper orthogonal decomposition (POD) technique may be used to reduce the unknowns in the CNFE method, which can be used to reduce many numerical methods (see, e.g., [17][18][19][20][21][22]). Our future work is reducing the number of unknowns in the CNFE method by the POD technique.…”

A reduced‐order extrapolating space–time continuous finite element method for the 2D Sobolev equation

“…The proper orthogonal decomposition (POD) method [11, 12] is an effective reduced‐order technique, which has been applied to various fields such as pattern recognition and data processing as well as statistics [13], geophysical fluid mechanics [14], and biomedical science [15]. Particularly, it plays an important role in the reduced‐order to the classical numerical methods, including the spectral methods, the natural boundary element methods, the finite difference (FD) schemes, the FE methods, the reduced basis methods, and the optimal control problems for PDEs [16–31].…”

A spectral element Crank–Nicolson model to the 2D unsteady conduction–convection problems about vorticity and stream functions

“…It is universally acknowledged that the spectral and finite element (FE) together with finite difference (FD) along with finite volume element (FVE) methods are four welcome numerical means (see [5][6][7][8][9][10]). Nevertheless, the spectral method possesses the highest precision among four numerical ones because the unknowns to the spectral method are approximated with the smooth functions, including trigonometric functions or the Chebyshev, Jacobi, and Legendre polynomials, but the unknowns to the FE and FVE methods are usually approximated by the classic polynomials, while the derivatives to the FD method are approached with difference quotients.…”

“…Specially, it is able to avoid the restriction for Babuska-Brezzi's stability conditions to spectral subspaces so as to be able to easily seek the SECN solutions, which is different from the previous other SE methods as stated above. As a consequence, the SECN model is fully distinguished from the spectral ones (see [8][9][10][11][12][13][14][15][16][17][18][19][20][21]) and is a development or a supplement to the previous ones.…”