In this paper, a family of high order numerical methods are designed to solve the Hamilton-Jacobi equation for the viscosity solution. In particular, the methods start with a hyperbolic conservation law system closely related to the Hamilton-Jacobi equation. The compact one-step one-stage Lax-Wendroff type time discretization is then applied together with the local-structure-preserving disconti...

For solving time-dependent convection-dominated partial differential equations (PDEs), which arise frequently in computational physics, high order numerical methods, including finite difference, finite volume, finite element and spectral methods, have been undergoing rapid developments over the past decades. In this article we give a brief survey of two selected classes of high order methods, n...

High order fast sweeping methods have been developed recently in the literature to solve static Hamilton-Jacobi equations efficiently. Comparing with the first order fast sweeping methods, the high order fast sweeping methods are more accurate, but they often require additional numerical boundary treatment for several grid points near the boundary because of the wider numerical stencil. It is p...

To gain insight into cardio-arterial interactions, a coupled left ventricle-systemic artery (LV-SA) model is developed that incorporates a three-dimensional finite-strain left ventricle (LV), and a physiologically-based one-dimensional model for the systemic arteries (SA). The coupling of the LV model and the SA model is achieved by matching the pressure and the flow rate at the aortic root, i....

We study the Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion and with monotonically increasing initial data using the Riemann-Hilbert (RH) approach. The solution of the Cauchy problem, in the zero dispersion limit, is obtained using the steepest descent method for oscillatory Riemann-Hilbert problems. The asymptotic solution is completely described by a scalar func...

A general methodology is introduced to build conservative numerical models for fluid simulations based on segregated schemes, where mass, momentum, and energy equations are solved by different methods. It especially designed here developing new discretizations of the total equation adapted a thermal coupling with lattice Boltzmann method (LBM). The proposed linear equivalence standard entropy e...

We develop a theoretical tool to examine the properties of numerical schemes for advection equations. To magnify the defects of a scheme we consider a convection-reaction equation 0021-9 doi:10. q Th * Co E-m ut þ ðjujq=qÞx 1⁄4 u; u; x 2 R; t 2 Rþ; q > 1: It is shown that, if a numerical scheme for the advection part is performed with a splitting method, the intrinsic properties of the scheme a...