Related papers: A new class of finite difference methods: The zigz…
Time integration methods for solving initial value problems are an important component of many scientific and engineering simulations. Implicit time integrators are desirable for their stability properties, significantly relaxing…
A new explicit stabilized scheme of weak order one for stiff and ergodic stochastic differential equations (SDEs) is introduced. In the absence of noise, the new method coincides with the classical deterministic stabilized scheme (or…
In this work, in order to obtain higher-order schemes for solving forward backward stochastic differential equations, we adopt the high-order multi-step method in [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36(4) (2014),…
We build a simple and general class of finite difference schemes for first order Hamilton-Jacobi (HJ) Partial Differential Equations. These filtered schemes are convergent to the unique viscosity solution of the equation. The schemes are…
In this paper, we present a novel explicit second order scheme with one step for solving the forward backward stochastic differential equations, with the Crank-Nicolson method as a specific instance within our proposed framework. We first…
In this paper, an efficient algorithm is presented by the extrapolation technique to improve the accuracy of finite difference schemes for solving the fractional boundary value problems with non-smooth solution. Two popular finite…
We introduce a class of unconditionally energy stable, high order accurate schemes for gradient flows in a very general setting. The new schemes are a high order analogue of the minimizing movements approach for generating a time discrete…
Based on our recent results, in this paper, a compact finite difference scheme is derived for a time fractional differential equation subject to the Neumann boundary conditions. The proposed scheme is second order accurate in time and…
We study the numerical anisotropy existent in compact difference schemes as applied to hyperbolic partial differential equations, and propose an approach to reduce this error and to improve the stability restrictions based on a previous…
High-order finite volume and finite element methods offer impressive accuracy and cost efficiency when solving hyperbolic conservation laws with smooth solutions. However, if the solution contains discontinuities, these high-order methods…
The "fast iterative shrinkage-thresholding algorithm", a.k.a. FISTA, is one of the most well-known first-order optimisation scheme in the literature, as it achieves the worst-case $O(1/k^2)$ optimal convergence rate in terms of objective…
This paper is concerned with moving mesh finite difference solution of partial differential equations. It is known that mesh movement introduces an extra convection term and its numerical treatment has a significant impact on the stability…
The dynamical behavior of switched affine systems is known to be more intricate than that of the well-studied switched linear systems, essentially due to the existence of distinct equilibrium points for each subsystem. First, under…
A high-order well-balanced scheme for the Euler equations with gravitation is presented. The scheme is able to preserve a spatially high-order accurate discrete representation of a large class of hydrostatic equilibria. It is based on a…
In this paper, a high-order and fast numerical method is investigated for the time-fractional Black-Scholes equation. In order to deal with the typical weak initial singularities of the solution, we construct a finite difference scheme with…
A new class of non-monotone finite difference (FD) approximation methods for approximating solutions to non-degenerate stationary Hamilton-Jacobi problems with Dirichlet boundary conditions is proposed and analyzed. The new FD methods add a…
We introduce a class of high order accurate, semi-implicit Runge-Kutta schemes in the general setting of evolution equations that arise as gradient flow for a cost function, possibly with respect to an inner product that depends on the…
Let $z_{1},z_{2},...,z_{N}$ be a sequence of distinct grid points. A finite difference formula approximates the $m$-th derivative $f^{(m)}(0)$ as $\sum w_{k}f(z_{k})$, with $w_{k}$ being the weights. We derive an algorithm for finding the…
We introduce an efficient and accurate staggered-grid finite-difference (SGFD) method to solve the two-dimensional elastic wave equation. We use a coupled first-order stress-velocity formulation. In the standard implementation of SGFD…
Stationary periodic patterns are widespread in natural sciences, ranging from nano-scale electrochemical and amphiphilic systems to mesoscale fluid, chemical and biological media and to macro-scale vegetation and cloud patterns. Their…