Finite difference method

Abstract: Fractional finite difference methods are useful to solve the fractional differential equations. The aim of this article is to prove the stability and convergence of the fractional Euler method, the fractional Adams method and the high order methods based on the convolution formula by using the generalized discrete Gronwall inequality. Numerical experiments are also presented, which verify the theoretical analysis.

Abstract: We present a two-level finite difference scheme for the approximation of nonlinear parabolic equations. Discrete inner products and the lowest-order Raviart--Thomas approximating space are used in the expanded mixed method in order to develop the finite difference scheme. Analysis of the scheme is given assuming an implicit time discretization. In this two-level scheme, the full nonlinear problem is solved on a "coarse" grid of size H. The nonlinearities are expanded about the coarse grid solution and an appropriate interpolation operator is used to provide values of the coarse grid solution on the fine grid in terms of superconvergent node points. The resulting linear but nonsymmetric system is solved on a "fine" grid of size h. Some a priori error estimates are derived which show that the discrete L\infty(L2) and L2(H1) errors are $O(h^2 + H^{4-d/2} + \Delta t)$, where $d \geq 1$ is the spatial dimension.

Abstract: New finite difference methods using Cartesian grids are developed for elliptic interface problems with variable discontinuous coefficients, singular sources, and nonsmooth or even discontinuous solutions. The new finite difference schemes are constructed to satisfy the sign property of the discrete maximum principle using quadratic optimization techniques. The methods are shown to converge under certain conditions using comparison functions. The coefficient matrix of the resulting linear system of equations is an M-matrix and is coupled with a multigrid solver. Numerical examples are also provided to show the efficiency of the proposed methods.

Abstract: Frequency-domain waveform inversion is typically perfomed using frequency-domain finite-difference modelling techniques. In 3D, these methods face significant computational challenges that limit any application to full-scale seismic applications. An alternative approach is to use a 3D time-domain finite-difference method and extract the frequency-domain wavefield by computing the terms of a discrete Fourier transform at each time step. This method combines the computational efficiency of 3D time-domain modelling while permitting casting the inverse problem in the frequency domain.