Yanghong Huang

José A. Carrillo, Alina Chertock, Yanghong Huang

We propose a positivity preserving entropy decreasing finite volume scheme for nonlinear nonlocal equations with a gradient flow structure. These properties allow for accurate computations of stationary states and long-time asymptotics demonstrated by suitably chosen test cases in which these features of the scheme are essential. The proposed scheme is able to cope with non-smooth stationary states, different time scales including metastability, as well as concentrations and self-similar behavior induced by singular nonlocal kernels. We use the scheme to explore properties of these equations beyond their present theoretical knowledge.

Yanghong Huang, Adam Oberman

The fractional Laplacian $(-Δ)^{α/2}$ is the prototypical non-local elliptic operator. While analytical theory has been advanced and understood for some time, there remain many open problems in the numerical analysis of the operator. In this article, we study several different finite difference discretisations of the fractional Laplacian on uniform grids in one dimension that takes the same form. Many properties can be compared and summarised in this relatively simple setting, to tackle more important questions like the nonlocality, singularity and flat tails common in practical implementations. The accuracy and the asymptotic behaviours of the methods are also studied, together with treatment of the far field boundary conditions, providing a unified perspective on the further development of the scheme in higher dimensions.

Xiao Wang, Wenpeng Shang, Xiaofan Li et al.

Non-Gaussian Lévy noises are present in many models for understanding underlining principles of physics, finance, biology and more. In this work, we consider the Fokker-Planck equation(FPE) due to one-dimensional asymmetric Lévy motion, which is a nonlocal partial differential equation. We present an accurate numerical quadrature for the singular integrals in the nonlocal FPE and develop a fast summation method to reduce the order of the complexity from $O(J^2)$ to $O(J\log J)$ in one time-step, where $J$ is the number of unknowns. We also provide conditions under which the numerical schemes satisfy maximum principle. Our numerical method is validated by comparing with exact solutions for special cases. We also discuss the properties of the probability density functions and the effects of various factors on the solutions, including the stability index, the skewness parameter, the drift term, the Gaussian and non-Gaussian noises and the domain size.

Yanghong Huang, Xiao Wang

Several finite difference methods are proposed for the infinitesimal generator of 1D asymmetric $α$-stable Lévy motions, based on the fact that the operator becomes a multiplier in the spectral space. These methods take the general form of a discrete convolution, and the coefficients (or the weights) in the convolution are chosen to approximate the exact multiplier after appropriate transform. The accuracy and the associated advantages/disadvantages are also discussed, providing some guidance on the choice of the right scheme for practical problems, like in the calculation of mean exit time for random processes governed by general asymmetric $α$-stable motions.

Yanghong Huang, Adam Oberman

The fractional Laplacian $(-Δ)^{α/2}$ is a non-local operator which depends on the parameter $α$ and recovers the usual Laplacian as $α\to 2$. A numerical method for the fractional Laplacian is proposed, based on the singular integral representation for the operator. The method combines finite difference with numerical quadrature, to obtain a discrete convolution operator with positive weights. The accuracy of the method is shown to be $O(h^{3-α})$. Convergence of the method is proven. The treatment of far field boundary conditions using an asymptotic approximation to the integral is used to obtain an accurate method. Numerical experiments on known exact solutions validate the predicted convergence rates. Computational examples include exponentially and algebraically decaying solution with varying regularity. The generalization to nonlinear equations involving the operator is discussed: the obstacle problem for the fractional Laplacian is computed.