Abstract. We present a numerical method for the approximation of solutions for the class of stochastic differential equations driven by Brownian motions which induce stochastic variation in fixed directions. This class of equations arises naturally in the study of population processes and chemical reaction kinetics. We show that the method constructs paths that are second order accurate in the ...

In numerical solutions to hyperbolic partial differential equations in multidimensions, in addition to dispersion and dissipation errors, there is a grid-related error (referred to as isotropy error or numerical anisotropy) that affects the directional dependence of the wave propagation. Difference schemes are mostly analyzed and optimized in one dimension, wherein the anisotropy correction may...

This article describes the development of a high order finite volume method for the solution of transonic flows. The high order of accuracy is achieved by a reconstruction procedure similar to the weighted essentially non-oscillatory schemes (WENO). On the contrary to the WENO schemes, the weighted least square (WLSQR) scheme is easily extensible to the case of complex geometry.

The flux reconstruction (FR) methodology has proved to be an attractive approach to obtaining high-order solutions to hyperbolic partial differential equations. However, the utilization of somewhat arbitrarily defined correction polynomials in the application of these schemes, while adding some flexibility, detracts from their ease of implementation and computational efficiency. This paper desc...

A second-order difference scheme for a first-order elliptic system with discontinuous coefficients is derived and studied. This approximation can be viewed as an improvement of the well-known scheme with harmonic averaging of the coefficients for a second order elliptic equation, which is first-order accurate for the gradient of the solution. The numerical experiments confirm the second order c...

The solution for the eikonal equation with a point-source condition has an upwind singularity at the source point as the eikonal solution behaves like a distance function at and near the source. As such, the eikonal function is not differentiable at the source so that all formally high-order numerical schemes for the eikonal equation yield first-order convergence and relatively large errors. Th...

Based on a novel numerical flux involving jumps of even order derivatives of the numerical solution, a direct discontinuous Galerkin (DDG) method for diffusion problems was introduced in [H. Liu and J. Yan, SIAM J. Numer. Anal. 47(1) (2009), 475-698]. In this work, we show that higher order (k≥4) derivatives in the numerical flux can be avoided if some interface corrections are included in the ...

We modify a Galerkin method for nonlinear hyperbolic equations so that it becomes a simpler method of lines, which may be viewed as a collocation method. The high order of accuracy is preserved. We present a linear wave analysis of the scheme and discuss some aspects of nonlinear problems. Our numerical experiments indicate that the addition of a proper artificial viscosity makes the method com...

This paper considers a family of spatially discrete approximations, including boundary treatment, to initial boundary value problems in evolving bounded domains. The presented method is based on the Cartesian grid embedded Finite-Difference method, which was initially introduced by Abarbanel and Ditkowski (ICASE Report No. 96-8, 1996; and J. Comput. Phys. 133(2), 1997) and Ditkowski (Ph.D. thes...