OGS
ODE Solver Library

## Detailed Description

This ODE solver library has been designed with an implicit first-order quasilinear ODE in mind. However, it is in principle not restricted to such a kind of equation, but can be extended to also solve other equation types. In particular it is possible to introduce equation types that are no ODEs, but are nonlinear equations without time derivative terms.

The aim of the library's design is being able to formulate FEM processes without having to care which time discretization scheme and which nonlinear iteration method will be used to solve it.

The library offers different time discretization schemes, cf. the conceputal remarks on them, namely the forward and backward Euler and Crank-Nicolson methods and Backward Differentiation Formulas. The design follows Gear's method, which also underlies the DASSL algorithm of Petzold et al., cf. Differential-algebraic equations article, section Numerical methods/Direct discretization [6].

A rough overview over the interplay between the various parts of this library is given in the image below. Therein red symbols indicate which classes own which matrices or vectors (for the meaning of the different symbols refer to the documentation of the respective classes). The word own should not be taken too strict in the sense of C++ member ownership, although currently it is implemented as such; but in the future this implementation detail might change. Rather own means that the class is in charge of the respective matrix or vector, i.e., it can read from and write to it or pass it on to functions; or in other words: The class knows about the meaning of that matrix/vector. Note that only matrices and vectors that describe some proper state of the respective classes are shown; those storing only intermediate computations have been omitted from the image.

Interplay of the different parts of the ODE

solver library at the example of a first-order implicit quasilinear ODE." \image latex ode-solver-concept.pdf "Interplay of the different parts of the ODE solver library at the example of a first-order implicit quasilinear ODE."

In the ODE solver library the instances of some classes can work together with several instances of other classes throughout there lifetime, whereas they have a strict one-to-one correspondence to objects of some different type. Those relations are given in the following table.

Class 1 Class 2 Relation Remarks
TimeDiscretizedODESystem ODESystem 1:1 the ODESystem

represents part of the state of the TimeDiscretizedODESystem TimeDiscretizedODESystem | TimeDiscretization | 1:1 | analogous for the TimeDiscretization TimeDiscretizedODESystem | MatrixTranslator | 1:1 | analogous for the MatrixTranslator NonlinearSolver | NonlinearSystem | 1:n | a nonlinear solver can solve various equations, one after the other LinearSolver | NonlinearSolver | 1:n | various NonlinearSolver's can share the same LinearSolver

## Enumerations

enum class  NumLib::IterationResult : char { NumLib::SUCCESS , NumLib::FAILURE , NumLib::REPEAT_ITERATION }
Status flags telling the NonlinearSolver if an iteration succeeded. More...

enum class  NumLib::NonlinearSolverTag : bool { NumLib::Picard , NumLib::Newton }
Tag used to specify which nonlinear solver will be used. More...

enum class  NumLib::ODESystemTag : char { NumLib::FirstOrderImplicitQuasilinear , NumLib::NeumannBC }
Tag used to specify the type of ODE. More...

## Functions

template<ODESystemTag ODETag>
std::unique_ptr< MatrixTranslator< ODETag > > NumLib::createMatrixTranslator (TimeDiscretization const &timeDisc)

std::pair< std::unique_ptr< NonlinearSolverBase >, NonlinearSolverTagNumLib::createNonlinearSolver (GlobalLinearSolver &linear_solver, BaseLib::ConfigTree const &config)

## ◆ IterationResult

 enum NumLib::IterationResult : char
strong

Status flags telling the NonlinearSolver if an iteration succeeded.

Enumerator
SUCCESS
FAILURE
REPEAT_ITERATION

Definition at line 21 of file EquationSystem.h.

## ◆ NonlinearSolverTag

 enum NumLib::NonlinearSolverTag : bool
strong

Tag used to specify which nonlinear solver will be used.

Enumerator
Picard

Picard fixpoint iteration scheme

Newton

Newton-Raphson iteration scheme

Definition at line 19 of file Types.h.

20 {
21  Picard ,
22  Newton
23 };

## ◆ ODESystemTag

 enum NumLib::ODESystemTag : char
strong

Tag used to specify the type of ODE.

Enumerator
FirstOrderImplicitQuasilinear

First order implicit quasi-linear ODE

This is an ODE of the form $$M(x,t)\cdot \dot x + K(x,t) \cdot x - b(x,t) =: r(\dot x, x, t) \stackrel{!}{=} 0$$

NeumannBC

Definition at line 26 of file Types.h.

27 {
35  NeumannBC // Sure, that's misuse of this enum, so sue me!
36 };

## ◆ createMatrixTranslator()

template<ODESystemTag ODETag>
 std::unique_ptr > NumLib::createMatrixTranslator ( TimeDiscretization const & timeDisc )

Creates a GlobalMatrix translator suitable to work together with the given time discretization scheme.

Definition at line 129 of file MatrixTranslator.h.

131 {
132  return std::unique_ptr<MatrixTranslator<ODETag>>(
133  new MatrixTranslatorGeneral<ODETag>(timeDisc));
134 }

## ◆ createNonlinearSolver()

 std::pair< std::unique_ptr< NonlinearSolverBase >, NonlinearSolverTag > NumLib::createNonlinearSolver ( GlobalLinearSolver & linear_solver, BaseLib::ConfigTree const & config )

Creates a new nonlinear solver from the given configuration.

Parameters
 linear_solver the linear solver that will be used by the nonlinear solver config configuration settings
Returns
a pair (nl_slv, tag) where nl_slv is the generated nonlinear solver instance and the tag indicates if it uses the Picard or Newton-Raphson method
Input File Parameter:
prj__nonlinear_solvers__nonlinear_solver__type
Input File Parameter:
prj__nonlinear_solvers__nonlinear_solver__max_iter
Input File Parameter:
prj__nonlinear_solvers__nonlinear_solver__damping
Input File Parameter:
prj__nonlinear_solvers__nonlinear_solver__prefix

Definition at line 408 of file NonlinearSolver.cpp.

410 {
412  auto const type = config.getConfigParameter<std::string>("type");
414  auto const max_iter = config.getConfigParameter<int>("max_iter");
415
416  if (type == "Picard")
417  {
418  auto const tag = NonlinearSolverTag::Picard;
419  using ConcreteNLS = NonlinearSolver<tag>;
420  return std::make_pair(
421  std::make_unique<ConcreteNLS>(linear_solver, max_iter), tag);
422  }
423  if (type == "Newton")
424  {
426  auto const damping = config.getConfigParameter<double>("damping", 1.0);
427  if (damping <= 0)
428  {
429  OGS_FATAL(
430  "The damping factor for the Newon method must be positive, got "
431  "{:g}.",
432  damping);
433  }
434  auto const tag = NonlinearSolverTag::Newton;
435  using ConcreteNLS = NonlinearSolver<tag>;
436  return std::make_pair(
437  std::make_unique<ConcreteNLS>(linear_solver, max_iter, damping),
438  tag);
439  }
440 #ifdef USE_PETSC
441  if (boost::iequals(type, "PETScSNES"))
442  {
443  auto prefix =
445  config.getConfigParameter<std::string>("prefix", "");
446  auto const tag = NonlinearSolverTag::Newton;
447  using ConcreteNLS = PETScNonlinearSolver;
448  return std::make_pair(std::make_unique<ConcreteNLS>(
449  linear_solver, max_iter, std::move(prefix)),
450  tag);
451  }
452
453 #endif
454  OGS_FATAL("Unsupported nonlinear solver type '{:s}'.", type.c_str());
455 }
#define OGS_FATAL(...)
Definition: Error.h:26

Referenced by ProjectData::parseNonlinearSolvers().