Difference Scheme For Diffusion Equation Research Articles

Backward differentiation formula finite difference schemes for diffusion equations with an obstacle term

Abstract Finite difference schemes, using backward differentiation formula (BDF), are studied for the approximation of one-dimensional diffusion equations with an obstacle term of the form $$\begin{equation*}\min(v_t - a(t,x) v_{xx} + b(t,x) v_x + r(t,x) v, v- \varphi(t,x))= f(t,x).\end{equation*}$$For the scheme building on the second-order BDF formula, we discuss unconditional stability, prove an $L^2$-error estimate and show numerically second-order convergence, in both space and time, unconditionally on the ratio of the mesh steps. In the analysis an equivalence of the obstacle equation with a Hamilton–Jacobi–Bellman equation is mentioned, and a Crank–Nicolson scheme is tested in this context. Two academic problems for parabolic equations with an obstacle term with explicit solutions and the American option problem in mathematical finance are used for numerical tests.

Stability Analysis of the Inverse Lax–Wendroff Boundary Treatment for High Order Central Difference Schemes for Diffusion Equations

In this paper, high order central finite difference schemes in a finite interval are analyzed for the diffusion equation. Boundary conditions of the initial-boundary value problem are treated by the simplified inverse Lax–Wendroff procedure. For the fully discrete case, a third order explicit Runge–Kutta method is used as an example for the analysis. Stability is analyzed by both the Gustafsson, Kreiss and Sundstrom theory and the eigenvalue visualization method on both semi-discrete and fully discrete schemes. The two different analysis techniques yield consistent results. Numerical tests are performed to demonstrate and validate the analysis results.

Convergence analysis of compact difference schemes for diffusion equation with nonlocal boundary conditions

Compact difference schemes for solving the diffusion equation with nonlocal boundary conditions are considered in this paper. Fourth-order compact difference is used to approximate the second order spatial derivative, and the integrals in the boundary conditions are approximated by the composite Simpson quadrature formula. The backward Euler and Crank–Nicolson schemes are presented as the fully discrete schemes. Error estimates in the discrete h1 and l∞ norms are given by the energy method, showing both schemes are fourth-order accurate in space, and they have first-order and second-order accuracy in time, respectively. Numerical results are provided to confirm the theoretical results.

Finite Difference Schemes for Diffusion Problems Based on a Hybrid Perturbation–Galerkin Method

A new technique for the development of finite difference schemes for diffusion equations is presented. The model equations are the one space variable advection diffusion equation and the two space variable diffusion equation, each with Dirichlet boundary conditions. A two-step hybrid technique, which combines perturbation methods based on the parameter ρ=Δt∕(Δx)2 with the Galerkin method, provides a systematic way to develop new finite difference methods, referred to as hybrid equations. The main contributions of this paper include the (1) recovery of classical explicit or implicit finite difference schemes using only the perturbation terms; (2) development of new finite difference schemes, referred to as hybrid equations, which have better stability properties than the classical finite difference equations, permitting the use of larger values of the parameter ρ; and (3) higher order accurate methods, with either O((Δx)4) or O((Δx)6) truncation error, formed by convex linear combinations of the classical and hybrid equations. The solution of the hybrid finite difference equations requires only a tridiagonal equation solver and, hence, does not lead to excessive computational effort.

NUMERICAL SOLUTION FOR ADVECTION-DIFFUSION EQUATION WITH SPATIALLY VARIABLE COEFFICIENTS

ABSTRACT Presented herein is a solution for one-dimensional dispersion of conservative and nonconservative pollutants in an open channel with steady unpolluted lateral inflow uniformly distributed over the whole length of the channel. To account for the lateral inflow into the channel the velocity of flow is taken proportional to distance and dispersion coefficient proportional to square of velocity. Through transformation the governing equation is changed into constant coefficient equation. The cubic spline interpolation is employed for solution of advection equation and Crank Nicholson finite difference scheme for diffusion equation. The computed results are compared with analytical solution for continuous plane injection of the pollutant at the upstream boundary of the model.

A variational difference scheme for the one-dimensional diffusion equation

In this paper we construct a homogeneous variational difference scheme for the diffusion equation assuming its coefficients to be bounded and measurable; the order of convergence of the scheme is O(h2). We consider the boundary value problem $$\frac{d}{{dx}}\left( {K(x)\frac{{du}}{{dx}}} \right) - g(x)u = - \frac{{dF}}{{dx}},0< x< X$$ (1) subject to the boundary conditions $$u(0) = a,u(X) = b$$ (2) .