Techniques to accelerate solution of governing differential equations

The numerous methodologies have been developed in order to accelerate the solution of the Governing equations for stationary problems are as follows:
1. Local time-stepping
2. Enthalpy damping
3. Residual smoothing
4. Multi-grid
5. Preconditioning
The first three local time-stepping, enthalpy damping, and preconditioning works by modifying the system of the ordinary differential equations, while the other two techniques are based on the improvements of the solution process. All methods can be applied to both the explicit and the implicit time-stepping schemes, with the exception of the residual smoothing, which was developed especially for the explicit multistage schemes. These acceleration techniques are applicable also to the inner iteration of the dual-time stepping scheme.
Local time-stepping
In Local time-stepping, the largest possible time step for each control volume is used to integrate the discretized governing equations. The local time step can be calculated using various formulae. As a result, the convergence to the steady state is strongly accelerated, but the transient solution is no longer temporally accurate.
Enthalpy damping
There are certain cases, where the total enthalpy is constant in the whole flow field. Generally situation like these occurs for external flows governed by the Euler equations in absence of heat sources and external forces. Computational Effort can be reduced using this fact. The first possibility would be to prescribe the value of the total enthalpy in the flow domain. In consequence, the energy equation can be omitted from the Euler equations. This saves memory and CPU time. This Methodology was suggested by Jameson for the solution of the potential flow equation. It employs the difference between the total enthalpy H and its free stream value H, to define an additional forcing term. With this, the convergence to the steady state can be considerably accelerated. The enthalpy damping is conducted as an additional step after each update of the flow solution.
Residual smoothing
The maximum CFL number and the convergence properties of the explicit multistage time-stepping scheme can be influenced by optimizing the stage coefficients. Jameson and Baker introduced this technique with the aim to lend the explicit scheme an implicit character and hence to increase the maximum allowable CFL number.
A further purpose of the residual smoothing is a better damping of the high frequency error components of the residual. This is of particular importance for a successful application of the Multi-grid method.
The residual smoothing can be implemented in explicit, implicit or in mixed manner respectively. The residual smoothing is usually applied in each stage of the explicit time-stepping scheme. The previously computed residuals are replaced by the smoothed residual before the solution is updated.
Multi-grid
It is a very powerful acceleration technique which is based on the solution of the governing equations on a series of successively coarser grids. The solution updates from the coarse grid are then combined and added to the solution on the finest grid. The technique was originally developed by Brandt for elliptic partial differential equations and later applied to the Euler equations by Jameson. After that, the Multi-grid scheme was employed to solve the Navier-Stokes equations. The Multi-grid method can be implemented for both the explicit and the implicit time-stepping schemes. The goal of the current research is the significant improvement of the efficiency of multi-grid for hyperbolic flow problems.
The basic idea of the Multi-grid scheme is to employ coarse grids in order to drive the solution on the finest grid faster to steady-state. Two effects are utilized for this purpose:
* Larger time steps can be employed on the coarser grids (owing to a larger control volume) in conjunction with a reduced numerical effort. Since the work for determining a new solution is distributed mainly over the coarser grids, a more rapid convergence and a reduction of the computing time results.
* The majority of the explicit and implicit time-stepping and iterative schemes reduces efficiently mainly the high-frequency components of the solution error. The low-frequency components are usually only hardly damped. This results in a slow convergence to the steady state, after the initial phase (where the largest errors are eliminated) is over. The multi-grid scheme helps at this point -the low-frequency components on the finest grid becomes high-frequency components on the coarser grids and are successively damped. As a result, the entire error is very quickly reduced, and the convergence is significantly accelerated.
Thus, as we can see, the success of the Multi-grid scheme depends heavily on good damping of the high-frequency error components by the time-stepping or iterative scheme.
Ppreconditioning
In the low subsonic Mach number regime, when the magnitude of the flow velocity becomes small in comparison with the acoustic speed, the convective terms of the governing equations become stiff. The stiffness of the governing equations (when marching in time) is determined by the characteristic condition number. This number is defined as the ratio of the largest to the smallest eigenvalue. A large condition number CN (i.e. for M ~ 0) reduces the efficiency of wave propagation. It slows down the convergence to steady state. Furthermore the schemes for compressible flows have an amount of artificial dissipation which does not scale correctly for Mach numbers approaching zero. Thus, the accuracy of such spatial discretization suffers at low Mach numbers.
If the velocity in the entire flow field is low (M < 0.2), then the compressibility effects can be neglected and the incompressible equations can be utilized. The incompressible Navier-Stokes equations can be solved by the well-known pressure-based schemes. The other possibility is the application of the artificial compressibility (or pseudo-compressibility) method. However, there are flow cases like:
* High speed flows with large embedded regions of low velocity. For example - the subsonic flow upstream of a strongly converging nozzle.
* Low-speed flows which are compressible due to density changes induced by heat sources. This occurs for surface heat transfer or volumetric heat addition (combustion simulation).
* Problems, where compressible and incompressible flow at varying Mach numbers occur side by side. Such situation arises, for instance, in propulsion, for high-lift configurations and in V/STOL maneuvering.
The above cases require the application of the compressible governing equations. In order to solve them preconditioning employed. The advantage of preconditioning is that it enables a solution method, which is applicable at all Math numbers. In the following, we shall derive the preconditioned governing equations.

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