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A three-step wavelet Galerkin method for parabolic and hyperbolic partial differential equations. (English)
Int. J. Comput. Math. 83, No. 1, 143-157 (2006).
Summary: A three-step wavelet Galerkin method based on Taylor series expansion in time is proposed. The scheme is third-order accurate in time and $O(2^{-jp})$ accurate in space. Unlike Taylor-Galerkin methods, the present scheme does not contain any new higher-order derivatives which makes it suitable for solving nonlinear problems. The compactly supported orthogonal wavelet bases $D6$ developed by {\it I. Daubechies} [Commun. Pure Appl. Math. 41, No.~7, 909‒996 (1988; Zbl 0644.42026)] are used in the Galerkin scheme. The proposed scheme is tested with both parabolic and hyperbolic partial differential equations. The numerical results indicate the versatility and effectiveness of the proposed scheme.