Abstract
In this paper, we shall be concerned with the numerical solution of parabolic equations in one space variable and the time variable t. We expand Taylor series to derive a higher-order approximation for Ut. We begin with the simplest model problem, for heat conduction in a uniform medium. For this model problem, an explicit difference method is very straightforward in use, and the analysis of its error is easily accomplished by the use of a maximum principle. As we shall show, however, the numerical solution becomes unstable unless the time step is severely restricted, so we shall go on to consider other, more elaborate, numerical methods which can avoid such a restriction. The additional complication in the numerical calculation is more than offset by the smaller number of time steps needed. We then extend the methods to problems with more general boundary conditions, then to more general linear parabolic equations. Finally, we shall discuss the more difficult problem of the solution of nonlinear equations.
Highlights
We shall be concerned with the numerical solution of parabolic equations in one space variable and the time variable t
We begin with the simplest model problem, for heat conduction in a uniform medium
An explicit difference method is very straightforward in use, and the analysis of its error is accomplished by the use of a maximum principle
Summary
Partial differential equations (PDEs) form the basis of very many mathematical models of physical, chemical and biological phenomena, and more recently their use has spread into economics, financial forecasting, image processing and other. In calculating the quantities to a good approximation, there is a thin boundary layer near the wing surface where viscous forces are important and that outside this an inviscid flow can be assumed. Where u is the flow velocity in the direction of the tangential co-ordinate x, y is the normal co-ordinate, ν is the viscosity, ρ is the density and p the pressure; we have here neglected the normal velocity. This is a typical parabolic equation for 1 ∂p u with ρ ∂x treated as a forcing term [3] [4]. Is the sound speed [5] [6]
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