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Add Generalized Extrapolation Method (GEM) time integration schemes

Merged Jacques Xing requested to merge CFD-Xing/nektar:feature/gem-time-integration-stand-alone into master

Issue/feature addressed

Add explicit and implicit extrapolation time-integration method. Extrapolation methods allow the construction of arbitrarily high-order time-integration methods based on a series for low-order approximations. Interested reader can look at the following references:

https://epubs.siam.org/doi/pdf/10.1137/1027140
https://epubs.siam.org/doi/epdf/10.1137/080732833
https://msp.org/camcos/2014/9-2/camcos-v9-n2-p01-p.pdf

Tests

The following test files have been added to the CI

library/Demos/LibUtilities/Tests/TimeIntegrationDemoGEMExplicitOrder1.tst
library/Demos/LibUtilities/Tests/TimeIntegrationDemoGEMExplicitOrder2.tst
library/Demos/LibUtilities/Tests/TimeIntegrationDemoGEMExplicitOrder3.tst
library/Demos/LibUtilities/Tests/TimeIntegrationDemoGEMExplicitOrder4.tst
library/Demos/LibUtilities/Tests/TimeIntegrationDemoGEMExplicitOrder5.tst
library/Demos/LibUtilities/Tests/TimeIntegrationDemoGEMExplicitMidpointOrder2.tst
library/Demos/LibUtilities/Tests/TimeIntegrationDemoGEMExplicitMidpointOrder4.tst
library/Demos/LibUtilities/Tests/TimeIntegrationDemoGEMExplicitMidpointOrder6.tst
library/Demos/LibUtilities/Tests/TimeIntegrationDemoGEMImplicitOrder1.tst
library/Demos/LibUtilities/Tests/TimeIntegrationDemoGEMImplicitOrder2.tst
library/Demos/LibUtilities/Tests/TimeIntegrationDemoGEMImplicitOrder3.tst
library/Demos/LibUtilities/Tests/TimeIntegrationDemoGEMImplicitOrder4.tst
library/Demos/LibUtilities/Tests/TimeIntegrationDemoGEMImplicitOrder5.tst
library/Demos/LibUtilities/Tests/TimeIntegrationDemoGEMImplicitMidpointOrder2.tst
library/Demos/LibUtilities/Tests/TimeIntegrationDemoGEMImplicitMidpointOrder4.tst
library/Demos/LibUtilities/Tests/TimeIntegrationDemoGEMImplicitMidpointOrder6.tst
library/Demos/LibUtilities/Tests/TimeIntegrationDemoGEMIMEXOrder1.tst
library/Demos/LibUtilities/Tests/TimeIntegrationDemoGEMIMEXOrder2.tst
library/Demos/LibUtilities/Tests/TimeIntegrationDemoGEMIMEXOrder3.tst
library/Demos/LibUtilities/Tests/TimeIntegrationDemoGEMIMEXOrder4.tst
library/Demos/LibUtilities/Tests/TimeIntegrationDemoGEMIMEXOrder5.tst

In addition, validation of the convergence of the numerical schemes have been realized

  1. The explicit Euler GEM scheme has been validated for order (1-6):
GEMEulerExplicitOrder1:
1.0002213408208864
GEMEulerExplicitOrder2:
2.000071255463732
GEMEulerExplicitOrder3:
3.0130960439559917
GEMEulerExplicitOrder4:
4.026899801460759
GEMEulerExplicitOrder5:
4.9584108709601775
GEMEulerExplicitOrder6:
5.971316749439216
  1. The explicit midpoint GEM scheme has been validated for order (2,4,6):
GEMMidpointExplicitOrder2:
2.000071255463732
GEMMidpointExplicitOrder4:
4.026899801460759
GEMMidpointExplicitOrder6:
5.971316749439216
  1. The implicit Euler GEM scheme has been validated for order (1-4):
GEMEulerImplicitOrder1:
0.9987680285674233
GEMEulerImplicitOrder2:
1.97898607303774
GEMEulerImplicitOrder3:
2.9412369884261667
GEMEulerImplicitOrder4:
3.8571772952155055
  1. The implicit "midpoint" GEM scheme has been validated for order (2,4,6):
GEMMidpointImplicitOrder2:
1.9941954378795832
GEMMidpointImplicitOrder4:
3.8664357920254795
GEMMidpointImplicitOrder6:
5.760570713650503
  1. The IMEX Euler GEM scheme has been implemented but its use is not recommended as the approach is affected by order-reduction problems.

Suggested reviewers

Please suggest any people who would be appropriate to review your code.

Notes

Checklist

  • Functions and classes, or changes to them, are documented.
  • User guide/documentation is updated.
  • Changelog is updated.
  • Suitable tests added for new functionality.
  • Contributed code is correctly formatted. (See the contributing guidelines).
  • License added to any new files.
  • No extraneous files have been added (e.g. compiler output or test data files).
Edited by Jacques Xing