A Survey of the Implementation of Numerical Schemes for Linear Advection Equation

Abstract

The interpolation method in a semi-Lagrangian scheme is decisive to its performance. Given the number of grid points one is considering to use for the interpolation, it does not necessarily follow that maximum formal accuracy should give the best results. For the advection equation, the driving force of this method is the method of the characteristics, which accounts for the flow of information in the model equation. This leads naturally to an interpolation problem since the foot point is not in general located on a grid point. We use another interpolation scheme that will allow achieving the high order for the box initial condition.

Numerical solution of the Laplace equation: electrostatic potential

“…The numerical solution is based on the finite difference method [6,7]. In the two-dimensional case we treat the plate as a mesh of discrete points.…”

“…The simplest case is that where the electric potential at the border is a fixed value, this type of condition is known as a Dirichlet boundary condition. Another type of condition is the Neumann boundary condition, which has as its data the derivative at the border [6,7].…”

A Special Case on the Stability and Accuracy for the 1D Heat Equation Using 3-Level and <i>θ</i>-Schemes

“…Many problems are modeled by smooth initial conditions and Dirichlet boundary conditions. A number of procedures have been suggested (see, for instance [1]- [3]). We can say that three classes of solution techniques have emerged for solution of PDE: the finite difference techniques, the finite element methods and the spectral techniques (see [4] and [5]).…”

A Special Case of Variational Formulation for Two-Point Boundary Value Problem in L2(Ω)

“…Much attention has been paid to the development, analysis and implementation of accurate methods for the numerical solution of this problem in the literature. Many problems are modeled by smooth initial conditions and Dirichlet boundary conditions, see [2] and [3]. We can say that three classes of solution techniques have emerged for solution of BVP for differential equations: the finite difference techniques, the finite element methods and the spectral techniques (see [4] and [5]).…”