It is well known that the spectral solutions of conservation laws have the attractive distinguishing property of infiniteorder convergence (also called spectral accuracy) when they are smooth (e.g., [C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods for Fluid Dynamics, Springer-Verlag, Heidelberg, 1988; J.P. Boyd, Chebyshev and Fourier Spectral Methods, second ed., Dover, New...

In this paper, we consider a variable-order fractional advection-diffusion equation with a nonlinear source term on a finite domain. Explicit and implicit Euler approximations for the equation are proposed. Stability and convergence of the methods are discussed. Moveover, we also present a fractional method of lines, a matrix transfer technique, and an extrapolation method for the equation. Som...

In this paper, we classified a class of fourth-order partial differential equations (PDEs) to be fourth-order PDE of type I, II, III and IV. The PDE of type IV is solved by using an efficient numerical method. The PDE is first transformed to a system of fourth-order ordinary differential equations (ODEs) using the method of lines, then the resulting system of fourth-order ODEs is solved using d...

When multilevel finite element spaces are not nested, different intergrid transfer operators would lead to different multigrid algorithms. It is proposed in this paper that discontinuous functions be averaged to continuous functions and that the bubble functions be discarded in the multigrid transferring. Applications of the techniques to various problems are presented with convergence analysis...

Babu ska and Yu constructed a posteriori estimates for nite element dis-cretization errors of linear elliptic problems utilizing a dichotomy principal stating that the errors of odd-order approximations arise near element edges as mesh spacing decreases while those of even-order approximations arise in element interiors. We construct similar a posteriori estimates for the spatial errors of nite...

Using a class of stochastic Euler and Runge-Kutta methods, we numerically solve a reaction-diffusion equation with additive random excitation. By discretizing the space and the associated stochastic differential system, we present a comparison of the diffusibility behaviors between the schemes above. The model presented here consists of reaction-diffusion equations describing the evolution of t...

In this paper an analysis will be presented for the ADI (alternating direction implicit) method of Peaceman and Rachford applied to initial-boundary value problems for partial differential equations in two space dimensions. We shall use the method of lines approach. Motivated by developments in the field of stiff nonlinear ordinary differential equations, our analysis will focus on problems whe...

We propose new discontinuous finite element methods that can be applied to one-dimensional elliptic problems with discontinuous coefficients. These methods are based on a class of higher degree immersed finite element spaces and can be used with a mesh independent of the location of coefficient discontinuity. Numerical experiments are presented to show that these methods can achieve optimal con...