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adeas31
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rewritten the section on integration methods
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‎UsersGuide/source/extrarefs.bib‎

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@@ -16,3 +16,61 @@ @BOOK{mathematicbook:third
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year = 1996
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}
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@book{Cellier:2006,
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author = {Cellier, Francois E. and Kofman, Ernesto},
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title = {Continuous System Simulation},
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year = {2006},
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isbn = {0387261028},
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publisher = {Springer-Verlag New York, Inc.},
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address = {Secaucus, NJ, USA},
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}
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@book{Hairer:1993,
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author = {Hairer, E. and N{\o}rsett, S. P. and Wanner, G.},
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title = {Solving Ordinary Differential Equations I: Nonstiff Problems},
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edition={2nd rev. ed. 1993. Corr. 3rd printing 2008},
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year = {1993},
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isbn = {978-3-540-56670-0},
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doi = {10.1007/978-3-540-78862-1},
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publisher = {Springer-Verlag Berlin Heidelberg},
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}
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@article{PetzoldDASSL:1982,
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title = {Description of DASSL: a differential/algebraic system solver},
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author = {Petzold, L.R.},
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journal = {},
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place = {United States},
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year = 1982,
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}
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@article{Hindmarsh:2005,
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title={{SUNDIALS}: Suite of nonlinear and differential/algebraic equation solvers},
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author={A.~C.~Hindmarsh and P.~N.~Brown and K.~E.~Grant and S.~L.~Lee and R.~Serban and D.~E.~Shumaker and C.~S.~Woodward},
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journal={ACM Transactions on Mathematical Software (TOMS)},
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volume={31},
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number={3},
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pages={363--396},
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year={2005},
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publisher={ACM}
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}
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@article{Davis:2010,
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author = {T.~A.~Davis and E.~Palamadai Natarajan},
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title = {Algorithm 907: KLU, A Direct Sparse Solver for Circuit Simulation Problems},
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journal = {ACM Trans. Math. Softw.},
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issue_date = {September 2010},
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volume = {37},
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number = {3},
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month = sep,
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year = {2010},
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issn = {0098-3500},
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pages = {36:1--36:17},
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articleno = {36},
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numpages = {17},
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url = {http://doi.acm.org/10.1145/1824801.1824814},
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doi = {10.1145/1824801.1824814},
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acmid = {1824814},
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publisher = {ACM},
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address = {New York, NY, USA},
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keywords = {LU factorization, circuit simulation, sparse matrices},
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}

‎UsersGuide/source/index.rst‎

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introduction
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omedit
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plotting
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solving
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debugger
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graphs
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fmi

‎UsersGuide/source/simulationflags.rst‎

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.. include :: simoptions.inc
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.. _cruntime-integration-methods :
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Integration Methods
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-------------------
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This section contains additional information about the different
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integration methods in OpenModelica, selected by the method flag
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of the :ref:`simulate` command or the :ref:`-s simflag <simflag-s>`.
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dassl
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~~~~~
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Default integration method in OpenModelica.
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Adams Moulton; the default uses a colored numerical Jacobian and interval root finding.
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To change settings, use simulation flags such as
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:ref:`dasslJacobian <simflag-dassljacobian>`,
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:ref:`dasslNoRootFinding <simflag-dasslnorootfinding>`,
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:ref:`dasslNoRestart <simflag-dasslnorestart>`,
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:ref:`initialStepSize <simflag-initialstepsize>`,
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:ref:`maxStepSize <simflag-maxstepsize>`,
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:ref:`maxIntegrationOrder <simflag-maxintegrationorder>`,
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:ref:`noEquidistantTimeGrid <simflag-noequidistanttimegrid>`.
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+----------------------+-----------------------------+
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| Order: | 1-5 |
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+----------------------+-----------------------------+
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| Step Size Control: | true |
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+----------------------+-----------------------------+
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| Order Control: | true |
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+----------------------+-----------------------------+
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| Stability Region: | variable; depend from order |
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+----------------------+-----------------------------+
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euler
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~~~~~
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Explicit Euler.
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+----------------------+---------------------+
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| Order: | 1 |
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+----------------------+---------------------+
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| Step Size Control: | false |
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+----------------------+---------------------+
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| Order Control: | false |
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+----------------------+---------------------+
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| Stability Region: | \|(1,0) Padé \| ≤ 1 |
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+----------------------+---------------------+
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rungekutta
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~~~~~~~~~~
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Classical Runge-Kutta method.
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+----------------------+---------------------+
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| Order: | 4 |
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+----------------------+---------------------+
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| Step Size Control: | false |
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+----------------------+---------------------+
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| Order Control: | false |
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+----------------------+---------------------+
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| Stability Region: | \|(4,0) Padé \| ≤ 1 |
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+----------------------+---------------------+
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radau1
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~~~~~~
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Radau IIA with one point.
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+----------------------+---------------------+
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| Order: | 1 |
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+----------------------+---------------------+
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| Step Size Control: | false |
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+----------------------+---------------------+
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| Order Control: | false |
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+----------------------+---------------------+
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| Stability Region: | \|(0,1) Padé \| ≤ 1 |
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+----------------------+---------------------+
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radau3
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~~~~~~
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Radau IIA with two points.
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+----------------------+---------------------+
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| Order: | 3 |
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+----------------------+---------------------+
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| Step Size Control: | false |
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+----------------------+---------------------+
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| Order Control: | false |
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+----------------------+---------------------+
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| Stability Region: | \|(1,2) Padé \| ≤ 1 |
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+----------------------+---------------------+
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radau5
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~~~~~~
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Radau IIA with three points.
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+----------------------+---------------------+
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| Order: | 5 |
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+----------------------+---------------------+
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| Step Size Control: | false |
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+----------------------+---------------------+
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| Order Control: | false |
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+----------------------+---------------------+
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| Stability Region: | \|(2,3) Padé \| ≤ 1 |
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+----------------------+---------------------+
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lobatto2
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~~~~~~~~
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Lobatto IIIA with two points.
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+----------------------+---------------------+
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| Order: | 2 |
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+----------------------+---------------------+
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| Step Size Control: | false |
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+----------------------+---------------------+
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| Order Control: | false |
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+----------------------+---------------------+
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| Stability Region: | \|(2,2) Padé \| ≤ 1 |
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+----------------------+---------------------+
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lobatto4
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~~~~~~~~
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Lobatto IIIA with three points.
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+----------------------+---------------------+
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| Order: | 4 |
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+----------------------+---------------------+
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| Step Size Control: | false |
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+----------------------+---------------------+
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| Order Control: | false |
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+----------------------+---------------------+
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| Stability Region: | \|(3,3) Padé \| ≤ 1 |
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+----------------------+---------------------+
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lobatto6
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~~~~~~~~
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Lobatto IIIA with four points.
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+----------------------+---------------------+
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| Order: | 6 |
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+----------------------+---------------------+
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| Step Size Control: | false |
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+----------------------+---------------------+
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| Order Control: | false |
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+----------------------+---------------------+
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| Stability Region: | \|(4,4) Padé \| ≤ 1 |
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+----------------------+---------------------+
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Notes
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~~~~~
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Simulation flags
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:ref:`maxStepSize <simflag-maxstepsize>` and
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:ref:`maxIntegrationOrder <simflag-maxintegrationorder>`
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specifiy maximum absolute step size and maximum integration order used by
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the dassl solver.
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General step size without control :math:`\approx \cfrac{\mbox{stopTime} - \mbox{startTime}}{\mbox{numberOfIntervals}}`.
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Events change the step size (see `Modelica spec 3.3 p. 88 <https://www.modelica.org/documents/ModelicaSpec33.pdf>`__).
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For (a,b) Padé see `wikipedia <http://en.wikipedia.org/wiki/Pad%C3%A9_table>`__.

‎UsersGuide/source/solving.rst‎

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.. _solving :
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Solving Modelica Models
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=======================
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.. TODO: Describe the Backend related modules.
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.. _cruntime-integration-methods :
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Integration Methods
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-------------------
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By default OpenModelica transforms a Modelica model into an ODE
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representation to perform a simulation by using numerical integration
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methods. This section contains additional information about the different
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integration methods in OpenModelica. They can be selected by the method
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parameter of the :ref:`simulate` command or the :ref:`-s simflag <simflag-s>`.
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The different methods are also called solver and can be distinguished by
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their characteristic:
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- explicit vs. implicit
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- order
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- step size control
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- multi step
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A good introduction on this topic may be found in :cite:`Cellier:2006`
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and a more mathematical approach can be found in :cite:`Hairer:1993`.
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.. _dassl :
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DASSL
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~~~~~
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DASSL is the default solver in OpenModelica, because of a severals reasons.
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It is an implicit, higher order, multi-step solver with a step-size control
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and with these properties it is quite stable for a wide range of models.
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Furthermore it has a mature source code, which was originally developed
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in the eighties an initial description may be found in :cite:`PetzoldDASSL:1982`.
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This solver is based on backward differentiation formula (BDF), which is
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a family of implicit methods for numerical integration. The used implementation
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is called DASPK2.0 (see [#f3]_) and it is translated automatically to C
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by f2c (see [#f4]_).
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The following simulation flags can be used to adjust the behavior of the
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solver for specific simulation problems:
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:ref:`jacobian <simflag-jacobian>`,
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:ref:`noRootFinding <simflag-norootfinding>`,
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:ref:`noRestart <simflag-norestart>`,
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:ref:`initialStepSize <simflag-initialstepsize>`,
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:ref:`maxStepSize <simflag-maxstepsize>`,
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:ref:`maxIntegrationOrder <simflag-maxintegrationorder>`,
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:ref:`noEquidistantTimeGrid <simflag-noequidistanttimegrid>`.
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IDA
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~~~
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The IDA solver is part of a software family called sundials: SUite of
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Nonlinear and DIfferential/ALgebraic equation Solvers :cite:`Hindmarsh:2005`.
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The implementation is based on DASPK with an extended linear solver
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interface, which includes an interface to the high performance sparse
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linear solver KLU :cite:`Davis:2010`.
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The simulation flags of :ref:`dassl` are also valid for the IDA
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solver and furthermore it has the following IDA specific flags:
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:ref:`idaLS <simflag-idaLS>`,
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:ref:`idaMaxNonLinIters <simflag-idaMaxNonLinIters>`,
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:ref:`idaMaxConvFails <simflag-idaMaxConvFails>`,
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:ref:`idaNonLinConvCoef <simflag-idaNonLinConvCoef>`,
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:ref:`idaMaxErrorTestFails <simflag-idaMaxErrorTestFails>`.
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Basic Explicit Solvers
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~~~~~~~~~~~~~~~~~~~~~~
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The basic explicit solvers are performing with a fixed step-size and
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differ only in the integration order. The step-size is based on the
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numberOfIntervals, the startTime and stopTime parameters in the
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:ref:`simulate` command:
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:math:`\mbox{stepSize} \approx \cfrac{\mbox{stopTime} - \mbox{startTime}}{\mbox{numberOfIntervals}}`
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- euler - order 1
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- heun - order 2
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- rungekutta - order 4
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Basic Implicit Solvers
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~~~~~~~~~~~~~~~~~~~~~~
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The basic implicit solvers are all based on the non-linear solver KINSOL
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from the SUNDIALS suite. The underlining linear solver can be modified
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with the simflag :ref:`-impRKLS <simflag-imprkls>`. The step-size is
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determined as for the basic explicit solvers.
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- impeuler - order 1
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- trapezoid - order 2
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- imprungekutta - Based on Radau IIA and Lobatto IIIA defined by its
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Butcher tableau where the order can be adjusted by :ref:`-impRKorder <simflag-imprkorder>`.
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Experimental Solvers
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~~~~~~~~~~~~~~~~~~~~
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The following solvers are marked as experimental, mostly because they
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are till now not tested very well.
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- rungekuttaSsc - Runge-Kutta based on Novikov (2016) - explicit, step-size control, order 4-5
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- irksco - Own developed Runge-Kutta solver - implicit, step-size control, order 1-2
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- symSolver - Symbolic inline solver (requires :ref:`--symSolver <omcflag-symSolver>`) - fixed step-size, order 1
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- symSolverSsc - Symbolic implicit inline Euler with step-size control (requires :ref:`--symSolver<omcflag-symSolver>`) - step-size control, order 1-2
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- qss - A QSS solver
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References
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~~~~~~~~~~
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.. bibliography:: openmodelica.bib extrarefs.bib
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:cited:
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:filter: docname in docnames
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.. rubric:: Footnotes
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.. [#f2] `Sundials Webpage <http://computation.llnl.gov/projects/sundials-suite-nonlinear-differential-algebraic-equation-solvers>`__
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.. [#f3] `DASPK Webpage <https://cse.cs.ucsb.edu/software>`__
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.. [#f4] 'Cdaskr source <https://github.com/wibraun/Cdaskr>`__

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