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Add Spectral Deferred Correction (SDC) time-stepping scheme

Jacques Xing requested to merge CFD-Xing/nektar:feature/sdc-time-integration into master

Issue/feature addressed

Add Spectral Deferred Correction (SDC) time-integration method. SDC 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://link.springer.com/content/pdf/10.1023/A:1022338906936.pdf
https://msp.org/camcos/2014/9-2/camcos-v9-n2-p01-p.pdf
https://arxiv.org/pdf/1706.06245.pdf
https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=73d1f8040544da722eac48398b851cb4ed5a06e8

The implemented approach include equidistant, Gauss-Lobatto-Legendre, Gauss-Radau-Legendre, Gauss-Lobatto-Chebyshev, and Gauss-Radau-Chebyshev quadrature combined with first-order explicit time-integration method and Gauss-Radau-Legendre quadrature combined with implicit time-integration method.

Tests

The following tests have been added to the CI:

TimeIntegrationDemoSDCExplicitEquidistantOrder2.tst
TimeIntegrationDemoSDCExplicitEquidistantOrder4.tst
TimeIntegrationDemoSDCExplicitEquidistantOrder6.tst
TimeIntegrationDemoSDCExplicitEquidistantOrder8.tst
TimeIntegrationDemoSDCExplicitGaussLobattoChebyshevOrder2.tst
TimeIntegrationDemoSDCExplicitGaussLobattoChebyshevOrder4.tst
TimeIntegrationDemoSDCExplicitGaussLobattoChebyshevOrder6.tst
TimeIntegrationDemoSDCExplicitGaussLobattoChebyshevOrder8.tst
TimeIntegrationDemoSDCExplicitGaussLobattoLegendreOrder2.tst
TimeIntegrationDemoSDCExplicitGaussLobattoLegendreOrder4.tst
TimeIntegrationDemoSDCExplicitGaussLobattoLegendreOrder6.tst
TimeIntegrationDemoSDCExplicitGaussLobattoLegendreOrder8.tst
TimeIntegrationDemoSDCImplicitGaussRadauLegendreOrder1.tst
TimeIntegrationDemoSDCImplicitGaussRadauLegendreOrder3.tst
TimeIntegrationDemoSDCImplicitGaussRadauLegendreOrder5.tst
TimeIntegrationDemoSDCImplicitGaussRadauLegendreOrder7.tst

In addition, the following convergence studies have been performed:

  1. The explicit SDC scheme has been validated with Equidistant quadrature nodes (2-5), corresponding to orders (2-5):
SDCExplicitOrder2:
2.0001849404258447
SDCExplicitOrder3:
2.998112275443462
SDCExplicitOrder4:
3.9072768262992614
SDCExplicitOrder5:
4.5995837300978915
  1. The explicit SDC scheme has been validated with GaussLobattoLegendre quadrature nodes (2-4), corresponding to orders (2,4,6):
SDCExplicitOrder2:
2.0000896766198726
SDCExplicitOrder4:
3.9723855124165213
SDCExplicitOrder6:
5.938139326221622
  1. The explicit SDC scheme has been validated with GaussRadauLegendre quadrature nodes (1-3), corresponding to orders (1,3,5):
SDCExplicitOrder1:
1.0069863770994856
SDCExplicitOrder3:
3.1122545359905596
SDCExplicitOrder5:
5.175226949995819
  1. The explicit SDC scheme has been validated with GaussLobattoChebyshev quadrature nodes (2-4), corresponding to orders (2,4,6):
SDCExplicitOrder2:
2.0001849404258447
SDCExplicitOrder4:
3.9851582816338422
SDCExplicitOrder6:
6.293402621379888
  1. The explicit SDC scheme has been validated with GaussRadauChebyshev quadrature nodes (2-4), corresponding to orders (2,4,6):
SDCExplicitOrder2:
2.0015302412770626
SDCExplicitOrder4:
4.164020377045453
SDCExplicitOrder6:
6.224349110295919
  1. The implicit SDC scheme has been validated with GaussRadauLegendre quadrature nodes (1-4), corresponding to orders (1,3,5,7):
SDCImplicitOrder1:
0.9972045679750912
SDCImplicitOrder3:
2.734499327650926
SDCImplicitOrder5:
4.598134392581629
SDCImplicitOrder7:
6.423925369611658
  1. The IMEX SDC 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

!1488 (merged) have to be merge first

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

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