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Numerical solution of parabolic equations in high dimensions

Tobias Von Petersdorff, Christoph Schwab (2004)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We consider the numerical solution of diffusion problems in ( 0 , T ) × Ω for Ω d and for T > 0 in dimension d 1 . We use a wavelet based sparse grid space discretization with mesh-width h and order p 1 , and h p discontinuous Galerkin time-discretization of order r = O ( log h ) on a geometric sequence of O ( log h ) many time steps. The linear systems in each time step are solved iteratively by O ( log h ) GMRES iterations with a wavelet preconditioner. We prove that this algorithm gives an L 2 ( Ω ) -error of O ( N - p ) for u ( x , T ) where N is the total number of operations,...

Numerical solution of parabolic equations in high dimensions

Tobias von Petersdorff, Christoph Schwab (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider the numerical solution of diffusion problems in (0,T) x Ω for Ω d and for T > 0 in dimension dd ≥ 1. We use a wavelet based sparse grid space discretization with mesh-width h and order pd ≥ 1, and hp discontinuous Galerkin time-discretization of order r = O ( log h ) on a geometric sequence of O ( log h ) many time steps. The linear systems in each time step are solved iteratively by O ( log h ) GMRES iterations with a wavelet preconditioner. We prove that this algorithm gives an L2(Ω)-error of O(N-p) for u(x,T)...

Numerical solution of second order one-dimensional linear hyperbolic equation using trigonometric wavelets

Mahmood Jokar, Mehrdad Lakestani (2012)

Kybernetika

A numerical technique is presented for the solution of second order one dimensional linear hyperbolic equation. This method uses the trigonometric wavelets. The method consists of expanding the required approximate solution as the elements of trigonometric wavelets. Using the operational matrix of derivative, we reduce the problem to a set of algebraic linear equations. Some numerical example is included to demonstrate the validity and applicability of the technique. The method produces very accurate...

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