The multi-scale finite element method is used to study the vibrational characteristics of polymer matrix reinforced by single-walled silicon carbide nanotubes. For this purpose, the nanoscale finite element method is employed to simulate the nanotubes at the nanoscale. While, the polymer is considered as a continuum at the larger scale. The polymer nanotube interphase is simulated by spring ele...

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...

where x = (x1, . . . , xd) ∈ IR , t > 0. HJ equations arise in many practical areas such as differential games, mathematical finance, image enhancement and front propagation. It is well known that solutions of (1) are Lipschitz continuous but derivatives can become discontinuous even if the initial data is smooth. There is a close relation between HJ equations and hyperbolic conservation laws. ...

A class of reconstructed discontinuous Galerkin (DG) methods is presented to solve compressible flow problems on arbitrary grids. The idea is to combine the efficiency of the reconstruction methods in finite volume methods and the accuracy of the DG methods to obtain a better numerical algorithm in computational fluid dynamics. The beauty of the resulting reconstructed discontinuous Galerkin (R...

We establish a general framework for analyzing the class of finite volume methods which employ continuous or totally discontinuous trial functions and piecewise constant test functions. Under the framework, optimal order convergence in the H1 and L2 norms can be obtained in a natural and systematic way for classical finite volume methods and new finite volume methods such as discontinuous finit...

Abstract. We present a numerical scheme for the solution of a class of atmospheric models where high horizontal resolution is required while a coarser vertical structure is allowed. The proposed scheme considers a layering procedure for the original set of equations, and the use of high-order ADER finite volume schemes for the solution of the system of balance laws arising from the dimensional ...

Utilization of symmetric condition in NASIR Galerkin Finite Volume Method for linear triangular element unstructured meshes is introduced for numerical solution of two dimensional strain and stress fields in a long thick cylinder section. The developed shape function free Galerkin Finite Volume structural solver explicitly computes stresses and displacements in Cartesian coordinate directions f...

We derive a posteriori error estimates for the discretization of the heat equation in a unified and fully discrete setting comprising the discontinuous Galerkin, finite volume, mixed finite element, and conforming and nonconforming finite element methods in space and the backward Euler scheme in time. Our estimates are based on a H-conforming reconstruction of the potential, continuous and piec...

The focus of this effort is to produce a two dimensional inviscid, compressible flow solver using the Discontinuous Galerkin Finite Element approach. The Discontinuous Galerkin method seeks to project the exact solution onto a finite polynomial space while allowing for discontinuities at cell interfaces. This allows for the natural discontinuity capture that is required for a compressible flow ...

We present an adaptive coupled level-set/volume-of-fluid (ACLSVOF) method for interfacial flow simulations on unstructured triangular grids. At each time step, we evolve both the level set function and the volume fraction. The level set function is evolved by solving the level set advection equation using a discontinuous Galerkin finite element method. The volume fraction advection is performed...