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- In this chapter we present the standard test cases for difference methods
for solving boundary-, initial-boundary and initial value problems
of the simplest partial differential equations.
- As a representative example of an elliptic boundary value problem we have
the Helmholtz equation
Δ u + λ × u = f
on a rectangle ( Δ = (d/dx)^2+(d/dy)^2 ) with
a possible mixture of Dirichlet (prescribed function value) or
von Neumann (presribed normal derivative value) boundary conditions.
This is solved using the simplest discretization scheme: the second order
derivative is replaced by the second order symmetric finite difference formula.
This results in the so called ''five point star'' combining 5 grid points
in the position of an axis parallel cross.
- The eigenvalue of the Laplace equation
Δ u = λ × u
with zero Dirichlet boundary conditions (simulating the clambed membrane)
is solved with the standard five-point star and the simutaneous inverse
iteration as eigenproblem solver.
- As the simplest example of an parabolic initial boundary value problem
we use the (simplified) model of thermal conduction in a insulated slab,
with given initial thermal values and prescribed temperature at the slabs ends.
This gives the problem
(d/dt)u = a (d/dx)^2 u + f(x,t)
where x in ]0,1[ and t > 0.
u(x,0), u(0,t) and u(1,t) are prescribed.
We have here the most used solution approach, the ''vertical methods of lines'',
where time is left nondiscretized first, the space variable x
replaced by a finite grid and the derivative (d/dx)^2 by the second order
symmetric finite difference formula with grid size δx.
This results in a system of coupled linear ordinary
differential equations. This system is stiff, the eigenvalus of the Jacobian vary
in the interval [-4/δx2,-π].
Subsequently it is fed into an ode-integrator which uses
an implicit multistep scheme, the backward differentiation formulae which are
especially good for this case.
- A strictly hyperbolic differential equation system of first order in two dimensions has two sets of
so called characteristic curves, which cover the plane two times. The method of
characteristics solves an initial value problem of such an equation on the
area of determination (a curved triangle) of a piece of the initial curve:
the initial curve is discretized by a grid. Through each grid point there pass two
such curves (one of each set). Each curve from the first set meets the curve from the
second set emanating from the neighboring point in a new grid point and the solution
values can be computed there. This gives a new grid on a new initial curve and the process
can be repeated. We present one example of this process with known solution.
- The convection equation
(d/dt)u + a (d/dx)u = 0 , a <> 0
is the simplest case of a hyperbolic PDE. We use this and present three finite difference schemes
for its numerical solution. (Since the exact solution is known, namely the shifted initial value
u(x-a*t,0)=u(x,t) this has no practical value, it serves simply for demonstrating
the effects of discretization error). The discretizations we use are
-
Friedrichs scheme (order 1)
- upwind scheme (order 1)
- Lax Wendroff (order 2 for smooth solutions)
- The wave equation
(d/dt)^2 u = c^2 (d/dx)^2 u , x in ]0,1[ , t>0
with given initial conditions on u(x,0) and (d/dt)u(x,0) as well as
prescribed values for u(0,t), u(1,t) describes the (undamped) vibration of
a string. This also is a well studied case, theoretically as well as numerically.
We use the classical 5 point star for solving this numerically.
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