Modified wavenumber

from Wikipedia, the free encyclopedia

In physics, the modified wave number is used to evaluate a discretization for a wave. Using the modified wave number, important numerical properties of a method, such as dispersion , numerical damping or the aliasing limit can be determined. It is also important for determining the time step limit in explicit numerical integration of time-dependent problems. The course of the modified wave number thus represents a more far-reaching evaluation measure than a pure order analysis. B. used in process development for direct numerical simulation .

Basics

A wave-based analysis is useful for evaluating a numerical method. On the one hand, a wave is a fundamental solution of almost all partial differential equations ; on the other hand, the wavelength provides a measure for the normalization of the step size. The quality of the method can thus be specified as a function of the points per wavelength. The simplest mathematical form of a wave is

with the imaginary unit of the wave number k . The first and second analytical derivation result in:

.

If one uses the wave number normalized with the step size

and solves the equations for the dimensionless wave number or its square, one obtains

.

The modified wave number results from the fact that the numerically determined derivatives are used in the above equations instead of the exact derivatives:

In the case of higher discharges, the procedure is carried out accordingly.

Dimensionless wave number

The range of the dimensionless wave number is limited to the value range 0 to . This corresponds to an arbitrarily fine resolution for or two points per wavelength for , which corresponds to the minimum required sampling rate according to the Nyquist frequency . The number of points per wavelength is the same . This relationship is shown in the following table for better clarity.

k * mod 0 π / 8 π / 4 π / 2 3/4 π π
Points / wavelength 16 8th 4th 2.66 2

Real and imaginary part of the modified wavenumber

The modified wavenumber is usually a complex quantity; H. In addition to the real part k * mod, r there can also exist an imaginary part k * mod, i . The real part is the amplitude of the calculated derivative of a wave and the imaginary part corresponds to the phase offset to the exact derivative. Since there is no preferred direction with a central discretization, there is no phase offset and thus the imaginary part of the modified wave number is zero. For higher derivatives, this applies accordingly to the square k * 2 mod, r and higher powers of the modified wavenumber. The influence of amplitude and phase errors of the numerically calculated derivatives is discussed in the sections First derivative and Second derivative .

example

In order to clearly explain the derivation of the modified wave number , the first derivative calculated with a central finite difference of the second order is considered as an example :

This gives the modified wave number to

.

Since this is a central difference, the modified wave number has only a real part and no imaginary part.

First derivative

The left diagram shows the modified wavenumber k * mod for various explicit and compact finite differences compared to the exact solution . Since these are central discretizations, the imaginary part k * mod, i is equal to zero and there is only one real part of k * mod . It can be clearly seen that for a higher order as well as when using a compact discretization scheme (red lines) the modified wavenumber for larger wavenumbers k * , i.e. H. for a lower resolution the exact solution follows.

The properties of various one-sided finite differences are shown in the right figure. The real part of the modified wavenumber is given for the first, second and third order by the blue lines. A higher order leads to a better match in the range of small dimensionless wavenumbers, but in the range of poorer resolutions ( k * > 1.2) the values ​​for the modified wavenumber are clearly too high (see also the section on connection with the time step limit ). For differences directed to the right, the imaginary part of the modified wavenumber is given by the dashed red lines.

Modified wavenumber for various central finite differences from second to sixth order.
Modified wavenumber for one-sided finite differences weighted to the right.

Fanning and damping

For the simple advection equation

with a positive advection speed c , a positive value of k * mod, i corresponds to an amplification. Because of the weighting of the difference star in the downstream direction, one speaks of a downwind method. In the case of an upwind method, only the sign of the imaginary part changes. This results in an almost continuously negative imaginary part that has a dampening effect and is therefore able to stabilize a numerical method.

Dispersion and Aliasing

For a transport equation there is an amplitude error, i. H. a deviation of k * mod, r from the exact solution k * , an error in the speed of propagation of a wave, which is called dispersion . From the advection equation it can be deduced that the phase velocity c ph is proportional to the ratio of the modified wave number k * mod, r and the dimensionless wave number k * . The group velocity c gr behaves like the derivative of the modified wave number:

  • Phase velocity:
  • Group speed:

Since the real part of the modified wave number goes to zero for the minimum resolution of two points per wavelength , phase velocities result from the maximum of k * mod, r that are already obtained for finer resolutions. The aliasing limit is thus the maximum of the modified wavenumber. There one obtains a group speed of zero and for larger values ​​of k * the result is a negative group speed, i.e. H. Disturbances spread against the physical direction of propagation, which is given by the partial differential equation to be solved . This also explains why a stretching to infinity (e.g. for boundary conditions), which can be mathematically formulated, is not suitable for unsteady problems.

Second derivative

Squares of the modified wavenumber for the second derivative, calculated with central finite differences of second to sixth order.

The diagram opposite shows the square of the modified wavenumber for various explicit and compact finite differences. Since these are central discretization stars, this has only one real part. It can be seen that with a higher order and with a compact discretization there is a better match with the exact solution k * 2 . In addition, the higher the quality of the discretization, the higher the value for the wave resolved with only two points. This means that poorly resolved waves are better attenuated due to the viscous terms, which is particularly important for non-linear calculations.

The real part k * 2 mod, r corresponds to the damping for the simple diffusion equation

.

In general, viscous terms, i.e. H. 2. Derivation not calculated with a skewed discretization, since the associated imaginary part k * 2 mod, i causes an artificial advection.

Connection with the order of consistency

In the consistency order, the behavior of the termination error is considered when the step size is reduced. For example, halving the step size for a second order discretization leads to a reduction of the error by a factor of 4. Since the consistency order is based on a Taylor series , this naturally only applies to small step sizes, i.e. H. for problems that are already well resolved. This is the case for k * → 0. The order therefore does not provide any information about the behavior of the numerical method for problems with poorer resolution. With a higher order there is a better match of the derivatives for k * = 0 with the exact solution, but the consistency order cannot be read off directly.

Relation to the time step limit

The time step limit for explicit integration methods is determined by the maximum frequencies that occur or the temporal amplification and attenuation rates. In the case of the advection equation, the greatest possible time derivative and thus the highest frequency is proportional to the maximum value of the first spatial derivative. The maximum value of the modified wave number k * mod is therefore a direct measure of the time step limit. In the case of diffusion problems, the greatest possible frequency scales with the maximum of the square of the modified wave number, so that here values ​​of k * 2 mod represent a greater restriction for the maximum possible time step.

See also

literature

  • S. Lele: Compact finite differences with spectral-like resolution. Journal of Computational Physics, Vol. 103, 1992, pp. 16-42, ISSN  0021-9991
  • M. Kloker: A robust high-resolution split-type compact FD scheme for spatial DNS of boundary-layer transition. Applied Scientific Research, Vol. 59, 1998, pp. 353-377, doi : 10.1023 / A: 1001122829539

Web links

  • Diagram catalog for the solution behavior of finite differences, lecture material University of Stuttgart (PDF file; 2.29 MB)