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title	Background:

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Let's test some of the ProjectX solvers for the case of a NACA0012

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airfoil in compressible, inviscid flow.

Each test case starts with an initial primal solution (sample x-momentum shown below). After reading the input, ProjectX runs a particular test case. Subsequently, all numerical data stored in the history (.hist) files are compared with a "true" history file.

In some cases, the solvers are also run in parallel using MPI and compared with the solvers run in serial. The drag and lift coefficients along with residual convergence are given.

For more information, please consult the .job and .hist files located in the corner of each panel.

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title	Inputs:

Common flow parameters:
- Equation set: Compressible 2D Euler
- Freestream Mach number: M _∞ = 0.5
- Angle of attack: α = 0 °

Total pressure: P_t =

Total temperature: T_t =

Static temperature:

- CFL: 10³⁰
Common solver parameters:
- Preconditioning:
  - Outer GMRES iterations (no. restarts): 20
  - Inner GMRES iterations (no. Krylov vectors): 200
- Solution order: 2
- PSequencing: False

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title	Boundary conditions:

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title	Input Primal X-Momentum:

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Linear solver tests:

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title	What is being tested?

Listed below are the tolerances for the relevant data compared in the history files:

Solution update fraction: 0.1
Residaul norm: 10^-10
Drag coefficient: 10^-12
Lift coefficient: 10^-12

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title	Residual convergence comparison:

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title	Newton-Jacobi (Left):

SolversNonlinear solver:
- NewtonNonlinear
Linear solver:
- PreconditionerProconditioner: Block Jacobi (left)
- Side: Left
Serial resultsResults:
- Residual norm converges to 4.36·10^3571·10^-13^ ¹³ after 5 iterations
- Drag coefficient: C_D = .53 5318 counts
- Lift coefficient: C_L = .96 9638 counts

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title	Newton-Jacobi (Right):

SolversNonlinear solver:
- NewtonNonlinear
Linear solver:
- Preconditioner: Block Jacobi (right)
- Side: Right
Results:
- Residual norm converges to 4.35·10^3532·10^-13^ ¹³ after 5 iterations
- Drag coefficient: C_D = .53 5318 counts
- Lift coefficient: C_L = .96 9638 counts

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Solvers

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title	Newton-ILU

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Nonlinear solver:
- Newton
Linear solver:
- Preconditioner: Incomplete LU (0) (left)
- Reordering type: Minimum discarded fill
- Side: Left
Serial resultsResults:
- Residual norm converges to 4.357·10^3566·10^-13^ ¹³ after 5 iterations
- Drag coefficient: C_D = .5318 counts
- Lift coefficient: C_L = .9638 counts

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Parallel results:

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title	Newton-ILU (Parallel)

Solvers:
- Nonlinear: Newton
- Preconditioner: Jacobi (right)
Results:
- Residual norm converges to 4.36·10^3501·10^-13^ ¹³ after 5 iterations
- Drag coefficient: C_D = .5318 counts
- Lift coefficient: C_L = .9638 counts

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Solvers

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title	Newton-ILUT

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Nonlinear solver:
- Newton
Linear solver:
- Preconditioner:Threshold-based Incomplete LU (ILUT) (right)
- Reordering type: Block Minimum Discarded Fill (MDF) Minimum discarded fill
- Side: RightFill level: 2
- Drop threshold: 10^-3^ 10^-3
Serial results:
- Residual norm converges to 4.3538·10^-13 after 5 iterations
- Drag coefficient: C_D = .5318 counts
- Lift coefficient: C_L = .9638 counts
Parallel resultsResults:
- Residual norm converges to 4.36·10^3539·10^-13^ ¹³ after 5 iterations
- Drag coefficient: C_D = .53 5318 counts
- Lift coefficient: C_L = .96 9638 counts

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Solvers

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title	Newton-UMFPACK

ILU Multigrid:

Nonlinear solver:
- Newton
- Preconditioner: UMFPACK (right)
Linear solver:
- Incomplete LU (0) (left) preconditioner
  - Reordering type: Minimum discarded fill
  - Side: Right
- P = 0 linear multigrid for coarse correction
Serial results:
- Residual norm converges to 3.8091·10^-13 after 5 iterations
- Drag coefficient: C_D = .5318 counts
- Lift coefficient: C_L = .9638 counts
Parallel resultsResults:
- Residual norm converges to 4.36·10^3501·10^-13^ ¹³ after 5 iterations
- Drag coefficient: C_D = .5318 counts
- Lift coefficient: C_L = .9638 counts

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Solvers

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title	Newton-ILU-ASM -MG (Parallel)

(with/without multigrid):

Nonlinear solver:
- Newton
Linear solver:
- Preconditioner: Jacobi (left): Incomplete LU (ILU)
- Parallel preconditioner: Additive Schwarz Method (ASM)
- Side: Right
- Reordering type: Minimum discarded fill
- Outer GMRES iterations: 2
- Inner GMRES iterations: 20
Parallel results (without multigrid)Results:
- Residual norm converges to 4.36·10^-13^ .3736·10^-13 after 5 iterations
- Drag coefficient: C_D = .5318 counts
- Lift coefficient: C_L = .9638 counts
Parallel results (with multigrid):
- Residual norm converges to 3.8106·10^-13 after 5 iterations
- Drag coefficient: C_D = .53 5318 counts
- Lift coefficient: C_L = .96 9638 counts

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Solvers

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title	Newton-ILU-ASM (Parallel)

UMFPACK:

Nonlinear solver:
- Newton
Linear solver:
- Preconditioner: Jacobi (right) Threshold-based Incomplete LU
- Side: Right
- Reordering type: Minimum discarded fill
Serial resultsResults:
- Residual norm converges to 4.36·10^3559·10^-13^ ¹³ after 5 iterations
- Drag coefficient: C_D = .5318 counts
- Lift coefficient: C_L = .9638 counts

- counts
Parallel results:
- Residual norm converges to 4.3506·10^-13 after 5 iterations
- Drag coefficient: C_D = .5318 counts
- Lift coefficient: C_L = .9638 counts

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title	Lean Jacobi:

SolversNonlinear solver:
- Lean JacobiNonlinear
Linear solver:
- Preconditioner: Jacobi (left) None
Serial results:
- Residual norm converges to 4.7345·10^-4 after 200 iterations
- Drag coefficient: C_D = .5098 counts
- Lift coefficient: C_L = .9478 counts
Parallel resultsResults:
- Residual norm converges to 4.36·10^7345·10^-13^ ⁴ after 5 200 iterations
- Drag coefficient: C_D = .53 5098 counts
- Lift coefficient: C_L = .96 9478 counts

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Solvers

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title	Newton-Osher

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Nonlinear solver:
- Newton
Linear solver:
- Preconditioner: Jacobi (right): Threshold-based Incomplete LU
- Side: Left
- Reordering type: Minimum discarded fill
Serial resultsResults:
- Residual norm converges to 45.36·10^6482·10^-13^ ¹⁴ after 5 38 iterations
- Drag coefficient: C_D = .5318 5319 counts
- Lift coefficient: C_L = .9638 counts9653 counts

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Adjoint solver tests:

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title	What is being tested?

Listed below are the tolerances for the relevant data compared in the history files:

Adjoint residual: 10^-12

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title	Adjoint solution:

For the primal solution, please refer to the Newton-ILU case above.

Adjoint parameters:
- Adjoint solution computed using drag as output
- Sensitivity parameter: Angle of attack
- Forward sensitivities are computed (i.e. dC_D/dα with a tangent linearization)
Serial results:
- Adjoint residual norm converges to 1.8574·10^-15 after 2 iterations
- dC_D/dα (forward sensitivity): 1.6544·10^-3
- dC_D/dα (adjoint): 1.6544·10^-3

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title	Dual X-momentum:

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Adaptation tests:

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title	What is being tested?

Listed below are the tolerances for the relevant data compared in the history files:

Number of degrees of freedom on adapted meshes: 1
Error indicator: 10^-12
Error estimate: 10^-12
Drag: 10^-12

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title	Adaptation Test (boundary-curved):

For the primal solution, please refer to the Newton-ILU case above (with Nonlinear solver iterations limited to 50)

Adjoint parameters:
- Adjoint output: Drag
- Patch Reconstruction H1: False
- Hessian Scalar Type: -1
  (Mach number for Hessian evaluation)
- Nonlinear Smoother: Newton (solved to 10^-6)
- Adjoint Smoother residual converged to 10^-6
- Primal truth surrogate not fully converged
- Adjoint truth fully converged
Adaptation parameters:
- Output Adapted: Drag
- Anisotropy method: Hessian
- Curved re-meshing method: Boundary-curved
- Number of adaptation iterations: 2
- hp adaptation space
- Adaptation algorithm: Fixed-Fraction marking
  - 30% refining and 20% coarsening fractions
  - Target degree-of-freedom increase factor: 1.0 (constant DOF)
- Forward sensitivities are computed
  (i.e. dC_D/dα with a tangent linearization)
Serial results:
- Initial mesh:
  - DOF = 990
  - Error estimate = 3.9138·10^-4
  - Drag = 5.3180·10^-4
- First adaptation:
  - DOF = 1008
  - Error estimate = 1.9066·10^-4
  - Drag = 2.6942·10^-4
- Second adaptation:
  - DOF = 1026
  - Error estimate = 1.3617·10^-4
  - Drag = 2.0600·10^-4

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title	Adaptation Test (elasticity):

Almost all parameters are identical to the boundary-curved adaptation test. The only differing parameters are:
- Target degree-of-freedom increase factor: 1.25
- Curved re-meshing type: Elasticity
Serial results:
- Initial mesh:
  - DOF = 990
  - Error estimate = 3.9138·10^-4
  - Drag = 5.3180·10^-4
- First adaptation:
  - DOF = 1008
  - Error estimate = 1.8590·10^-4
  - Drag = 2.6109·10^-4
- Second adaptation:
  - DOF = 1278
  - Error estimate = 8.5377·10^-5
  - Drag = 1.3336·10^-4

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title	Primal X-momentum after two adaptation iterations (elasticity):

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title	Error estimate and drag over two adaptations (boundary-curved):

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title	Error estimate and drag over two adaptations (elasticity):

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Key

Linear solver tests:

Adjoint solver tests:

Adaptation tests: