In this paper, we develop the high order weighted essentially non-oscillatory (WENO) schemes for solving the Degasperis-Procesi (DP) equation, including finite volume (FV) and finite difference (FD) methods. The DP equation contains nonlinear high order derivatives, and possibly discontinuous or sharp transition solutions. The finite volume method is de- signed based on the total variation bounded property of the DP equation. And the finite difference method is constructed based on the L2 stability of the DP equation. Due to the adoption of the WENO reconstruction, both schemes are arbitrary high order accuracy and shock capturing. The numerical simulation results for different types of solutions of the nonlinear Degasperis-Procesi equation are provided to illustrate the accuracy and capability of the methods.

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Let M be a complete non-compact Riemannian manifold whose sectional curvature is bounded between two constants-k and K. Then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in Euclidean space. The purpose of this paper is to give a justi-fication of such a statement.

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This article studies the problem of image restoration of observed images corrupted by impulse noise and mixed Gaussian impulse noise. Since the pixels damaged by impulse noise contain no information about the true image, how to find this set correctly is a very important problem. We propose two methods based on blind inpainting and $\ell_0$ minimization that can simultaneously find the damaged pixels and restore the image. By iteratively restoring the image and updating the set of damaged pixels, these methods have better performance than other methods, as shown in the experiments. In addition, we provide convergence analysis for these methods; these algorithms will converge to coordinatewise minimum points. In addition, they will converge to local minimum points (or with probability one) with some modifications in the algorithms.