The following guide is an exemplary recipe to setup a basic simulation with the DG solver. For general settings (e.g. I/O), which are not specific to DG, see general application control. Properties in angle brackets '''< >''' indicate all possible choices. It is also encouraged to examine the testcases. The link to the repository can be found in the references.

Problem Definition

Activate the DG solver and set the dimension of the spatial discretization:

solvertype.default = "MAIA_UNIFIED"
solvertype.0 = "MAIA_DISCONTINUOUS_GALERKIN"
nDim = <2,3>

The DG solver supports space discretizations in 2D and 3D. The first step is to choose the right system of equations to solve

dgSystemEquations = <"DG_SYSEQN_ACOUSTICPERTURB", "DG_SYSEQN_LINEARSCALARADV">

For the properties specific to the available particular system of equations, continue here. The space discretization by the DG method and the Runge-Kutta time integration can be controlled separately and are outlined in the next section.

Space Discretization

The solver operates on a Cartesian mesh and supports local grid refinements, which in the DG terminology is called h-refinement. Additionally, the spatial resolution can be controlled by the choice of the basis function, which for the collocation type approach also controls the numerical quadrature

dgIntegrationMethod = <"DG_INTEGRATE_GAUSS", "DG_INTEGRATE_GAUSS_LOBATTO">

As a rule of thumb Gauss DGSEM has superior dispersion properties (cf. Gassner2011) and is preferred for acoustic application. DG_INTEGRATE_GAUSS_LOBATTO satisfies the summation-by-parts property (SBP) and might be superior in terms of numerical stability. One advantage of the DG method is the ability to locally control the spatial discretization order by adjusting the degree of the polynomial basis functions in each cell, called p-refinement

initPolyDeg       = 3
minPolyDeg        = 3
maxPolyDeg        = 4
pref              = 1                               // Activate p-refinement
prefCoordinates_0 = [0.5, 0.0, 0.0, 1.0, 1.0, 0.75] // Box p-refinement [-x,-y,-z,+x,+y,+z]
prefPolyDeg_0     = 4                               // Polynomial order of 0th patch

This example defines one p-refinement patch with index zero. Multiple p-refinement patches can be combined. Note that the time step \(\Delta t\) is a function of the polynomial degree of the basis functions, see here.

Time Discretization

The (temporal) integration method to be used in the DG solver can be set by

dgTimeIntegrationScheme = "DG_TIMEINTEGRATION_CARPENTER_4_5" // 4th order, 5 stages

cfl = 1.0

A complete list of all available integration schemes are listed here: dgTimeIntegrationScheme. The conversion from the cfl number to the time step \(\Delta t\) is scaled for all combinations of dgSystemEquations and basis functions to get for most applications the stability limit for cfl = 1.0. Note, that in general it is recommended to adapt the spatial and temporal order of accuracy. For coupled simulations an optimal interleaving of all solvers is achived with matching stage numbers of the Runge Kutta schemes.

System of Equations

The DG solver is designed for the solution of time dependent systems of first order convection dominated PDEs. Besides the common configuration of the DG solver described in the previous sections, depending on the underlying system of equations to be solved, further configuration is required in particular for the initial and boundary conditions.

APE

The acoustic perturbation equations in the APE-4 variant require the specification of mean variables, so called nodeVars, which are time independent in addition to the initial acoustic field, both set in DgSysEqnAcousticPerturb<nDim>::calcInitialCondition(). The type of initial condition can be controlled by the property initialCondition. A list of all possible values can be found here.

The acoustic waves can be excited by noise sources, which can be analytically described or taken from a CFD solution. The property sourceTerm selects the source formulation. A list of all possible values can be found here.

To fully define the physical problem proper boundary conditions must be specified in geometry.toml. A list of all possible boundary conditions can be found here.

Standalone

The APE-4 system can be solved with prescribed noise sources for benchmarking and testing new DG features.

Online Coupled

In the directly coupled setup the noise sources are exchanged in memory from a concurrently running CFD simulation. The mean quantities and the source terms in the APE-4 are extracted from the respective CFD simulation, and hence

initialCondition = 790 // nodeVars are not set & all fluctuations are set to zero

sourceTerm = 0

Todo:: Coupling related configuration

For more technical details about the coupling, see here.

Linear advection equation

A list of all possible values for initialCondition can be found here.
A list of all possible values for sourceTerm can be found here.

The theory to linear advection equations can be found here.

References

Details about the specific implementation of the DG scheme can be found here or in Schlottke2017.
A reference to the full list of properties can be found here.
A collection of testcases to start with can be found in the subversion repositories url for DG only and FV-DG coupled setups or url for LB-DG coupled setups.

Literature

Schlottke-Lakemper M. A Direct-Hybrid Method for Aeroacoustic Analysis. Dissertation, RWTH Aachen University, 2017, 10.18154/RWTH-2017-04082.
Gassner G, Kopriva D. A comparison of the dispersion and dissipation errors of Gauss and Gauss–Lobatto discontinuous Galerkin spectral element methods. SIAM Journal on Scientific Computing, 33(5), 2560-2579, 2011, 10.1137/100807211.

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