Finite difference methods ( FDM for short ), also methods of finite differences, are a class of numerical methods for solving ordinary and partial differential equations .
The basic idea of the procedure is to approximate the local derivatives in the differential equation at a finite number of (= "finite"), equidistant grid points using difference quotients. The approximated solutions of the differential equation at the grid points can then be calculated using the corresponding system of equations.
Processes of this type are widely used in fluid dynamic simulations , for example in meteorology and astrophysics . The difference method is used to a certain extent in structural engineering . As early as 1904, Friedrich Bleich analyzed the continuous beam; In 1909 Lewis Fry Richardson examined elastic disks and in 1919 Henri Marcus examined elastic plates using the difference method.
A special finite difference method for the numerical solution of the heat conduction equation is the Crank-Nicolson method .
Lewis Fry Richardson , Richard Southwell , Richard Courant , Kurt Friedrichs , Hans Lewy , Peter Lax and John von Neumann are among the pioneers of the finite difference method for partial differential equations .
Example for the numerical solution of an ordinary differential equation
The boundary value problem is given
The solution function can be calculated exactly here .
For the solution with the difference method, the interval is discretized by the grid points for with the mesh size . The second derivative is discretized using the central difference quotients of the second derivative
This gives the difference equations at the inner grid points
for the numerical approximate values of the solution values . Using the given boundary values and this is a linear system of equations with equations for the unknowns .
In matrix form , the system to be solved is here:
Since a maximum of three unknowns occur in each line, it is a system with a sparse coefficient matrix , more precisely a system with a tridiagonal Toeplitz matrix .
Example for the numerical solution of a partial differential equation
In the following, the numerical solution of the heat conduction equation is considered in a restricted area :
Numerical solution in 1D
In the 1D case there is a bounded interval. Since in this case only a spatial derivative is considered, the heat conduction equation can be written as follows:
In order to be able to use the finite difference method, the interval must first be divided into a finite number of sub-intervals. For this purpose, equidistant sampling points used:
, for .
The grid width of this discretization is therefore . According to the assumption, the function you are looking for disappears at the boundary values , i.e. H. so that these values do not have to be considered further. This allows the function evaluations to be displayed as a vector in the support points :
Approximation of the derivative
The second derivative of the location can now be approximated at the support points by second order difference quotients:
If the heat conduction equation is rearranged, the following system of ordinary first-order differential equations results:
where and .
This system can now by any method for solving ordinary differential equations, such as B. the Runge-Kutta method or the Euler method can be solved.
Goodness of the approximation
A finite difference method generates a linear system of equations (analogous to the equation in the example chapter )
where is the numerical approximation of the solution and should explicitly represent the dependence on the grid. Let be the exact solution and the finite representation using .
An FDM is called consistent of order , if there is one with
FDM is stable , if one is so for all true
One can show that convergence already follows from consistency and stability, that is
- Christian Großmann, Hans-Görg Roos: Numerical treatment of partial differential equations. 3. Edition. BG Teubner Verlag, Wiesbaden 2005, ISBN 3-519-22089-X .
- Stig Larsson, Vidar Thomée: Partial differential equations and numerical methods. Springer-Verlag, Berlin 2005, ISBN 3-540-20823-2 .
- Claus-Dieter Munz, Thomas Westermann: Numerical treatment of ordinary and partial differential equations. 3. Edition. Springer-Verlag, ISBN 978-3-642-24334-9