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

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.