# 3.4.2.3. Relaxation Methods

This section describes user-callable functions for applying relaxation methods with ARKStep. For more information on relaxation Runge–Kutta methods see §3.2.14.

Note

Relaxation support as not been evaluated with non-identity mass matrices. While this usage mode is supported, feedback from users who explore this combination would be appreciated.

## 3.4.2.3.1. Enabling or Disabling Relaxation

int ARKStepSetRelaxFn(void *arkode_mem, ARKRelaxFn rfn, ARKRelaxJacFn rjac)

Attaches the user supplied functions for evaluating the relaxation function (rfn) and its Jacobian (rjac).

Both rfn and rjac are required and an error will be returned if only one of the functions is NULL. If both rfn and rjac are NULL, relaxation is disabled.

With DIRK and IMEX-ARK methods or when a fixed mass matrix is present, applying relaxation requires allocating $$s$$ additional state vectors (where $$s$$ is the number of stages in the method).

Parameters:
• arkode_mem – the ARKStep memory structure

• rfn – the user-defined function to compute the relaxation function $$\xi(y)$$

• rjac – the user-defined function to compute the relaxation Jacobian $$\xi'(y)$$

Return values:
• ARK_SUCCESS – the function exited successfully

• ARK_MEM_NULLarkode_mem was NULL

• ARK_ILL_INPUT – an invalid input combination was provided (see the output error message for more details)

• ARK_MEM_FAIL – a memory allocation failed

Warning

Applying relaxation requires using a method of at least second order with $$b^E_i \geq 0$$ and $$b^I_i \geq 0$$. If these conditions are not satisfied, ARKStepEvolve() will return with an error during initialization.

Note

When combined with fixed time step sizes, ARKStep will attempt each step using the specified step size. If the step is successful, relaxation will be applied, effectively modifying the step size for the current step. If the step fails or applying relaxation fails, ARKStepEvolve() will return with an error.

## 3.4.2.3.2. Optional Input Functions

This section describes optional input functions used to control applying relaxation.

int ARKStepSetRelaxEtaFail(void *arkode_mem, sunrealtype eta_rf)

Sets the step size reduction factor applied after a failed relaxation application.

The default value is 0.25. Input values $$\leq 0$$ or $$\geq 1$$ will result in the default value being used.

Parameters:
• arkode_mem – the ARKStep memory structure

• eta_rf – the step size reduction factor

Return values:
• ARK_SUCCESS – the value was successfully set

• ARK_MEM_NULLarkode_mem was NULL

• ARK_RELAX_MEM_NULL – the internal relaxation memory structure was NULL

int ARKStepSetRelaxLowerBound(void *arkode_mem, sunrealtype lower)

Sets the smallest acceptable value for the relaxation parameter.

Values smaller than the lower bound will result in a failed relaxation application and the step will be repeated with a smaller step size (determined by ARKStepSetRelaxEtaFail()).

The default value is 0.8. Input values $$\leq 0$$ or $$\geq 1$$ will result in the default value being used.

Parameters:
• arkode_mem – the ARKStep memory structure

• lower – the relaxation parameter lower bound

Return values:
• ARK_SUCCESS – the value was successfully set

• ARK_MEM_NULLarkode_mem was NULL

• ARK_RELAX_MEM_NULL – the internal relaxation memory structure was NULL

int ARKStepSetRelaxUpperBound(void *arkode_mem, sunrealtype upper)

Sets the largest acceptable value for the relaxation parameter.

Values larger than the upper bound will result in a failed relaxation application and the step will be repeated with a smaller step size (determined by ARKStepSetRelaxEtaFail()).

The default value is 1.2. Input values $$\leq 1$$ will result in the default value being used.

Parameters:
• arkode_mem – the ARKStep memory structure

• upper – the relaxation parameter upper bound

Return values:
• ARK_SUCCESS – the value was successfully set

• ARK_MEM_NULLarkode_mem was NULL

• ARK_RELAX_MEM_NULL – the internal relaxation memory structure was NULL

int ARKStepSetRelaxMaxFails(void *arkode_mem, int max_fails)

Sets the maximum number of times applying relaxation can fail within a step attempt before the integration is halted with an error.

The default value is 10. Input values $$\leq 0$$ will result in the default value being used.

Parameters:
• arkode_mem – the ARKStep memory structure

• max_fails – the maximum number of failed relaxation applications allowed in a step

Return values:
• ARK_SUCCESS – the value was successfully set

• ARK_MEM_NULLarkode_mem was NULL

• ARK_RELAX_MEM_NULL – the internal relaxation memory structure was NULL

int ARKStepSetRelaxMaxIters(void *arkode_mem, int max_iters)

Sets the maximum number of nonlinear iterations allowed when solving for the relaxation parameter.

If the maximum number of iterations is reached before meeting the solve tolerance (determined by ARKStepSetRelaxResTol() and ARKStepSetRelaxTol()), the step will be repeated with a smaller step size (determined by ARKStepSetRelaxEtaFail()).

The default value is 10. Input values $$\leq 0$$ will result in the default value being used.

Parameters:
• arkode_mem – the ARKStep memory structure

• max_iters – the maximum number of solver iterations allowed

Return values:
• ARK_SUCCESS – the value was successfully set

• ARK_MEM_NULLarkode_mem was NULL

• ARK_RELAX_MEM_NULL – the internal relaxation memory structure was NULL

int ARKStepSetRelaxSolver(void *arkode_mem, ARKRelaxSolver solver)

Sets the nonlinear solver method used to compute the relaxation parameter.

The default value is ARK_RELAX_NEWTON.

Parameters:
• arkode_mem – the ARKStep memory structure

• solver – the nonlinear solver to use: ARK_RELAX_BRENT or ARK_RELAX_NEWTON

Return values:
• ARK_SUCCESS – the value was successfully set

• ARK_MEM_NULLarkode_mem was NULL

• ARK_RELAX_MEM_NULL – the internal relaxation memory structure was NULL

• ARK_ILL_INPUT – an invalid solver option was provided

int ARKStepSetRelaxResTol(void *arkode_mem, sunrealtype res_tol)

Sets the nonlinear solver residual tolerance to use when solving (3.48).

If the residual or iteration update tolerance (see ARKStepSetRelaxMaxIter()) is not reached within the maximum number of iterations (determined by ARKStepSetRelaxMaxIters()), the step will be repeated with a smaller step size (determined by ARKStepSetRelaxEtaFail()).

The default value is $$4 \epsilon$$ where $$\epsilon$$ is floating-point precision. Input values $$\leq 0$$ will result in the default value being used.

Parameters:
• arkode_mem – the ARKStep memory structure

• res_tol – the nonlinear solver residual tolerance to use

Return values:
• ARK_SUCCESS – the value was successfully set

• ARK_MEM_NULLarkode_mem was NULL

• ARK_RELAX_MEM_NULL – the internal relaxation memory structure was NULL

int ARKStepSetRelaxTol(void *arkode_mem, sunrealtype rel_tol, sunrealtype abs_tol)

Sets the nonlinear solver relative and absolute tolerance on changes in $$r$$ iterates when solving (3.48).

If the residual (see ARKStepSetRelaxResTol()) or iterate update tolerance is not reached within the maximum number of iterations (determined by ARKStepSetRelaxMaxIters()), the step will be repeated with a smaller step size (determined by ARKStepSetRelaxEtaFail()).

The default relative and absolute tolerances are $$4 \epsilon$$ and $$10^{-14}$$, respectively, where $$\epsilon$$ is floating-point precision. Input values $$\leq 0$$ will result in the default value being used.

Parameters:
• arkode_mem – the ARKStep memory structure

• rel_tol – the nonlinear solver relative solution tolerance to use

• abs_tol – the nonlinear solver absolute solution tolerance to use

Return values:
• ARK_SUCCESS – the value was successfully set

• ARK_MEM_NULLarkode_mem was NULL

• ARK_RELAX_MEM_NULL – the internal relaxation memory structure was NULL

## 3.4.2.3.3. Optional Output Functions

This section describes optional output functions used to retrieve information about the performance of the relaxation method.

int ARKStepGetNumRelaxFnEvals(void *arkode_mem, long int *r_evals)

Get the number of times the user’s relaxation function was evaluated.

Parameters:
• arkode_mem – the ARKStep memory structure

• r_evals – the number of relaxation function evaluations

Return values:
• ARK_SUCCESS – the value was successfully set

• ARK_MEM_NULLarkode_mem was NULL

• ARK_RELAX_MEM_NULL – the internal relaxation memory structure was NULL

int ARKStepGetNumRelaxJacEvals(void *arkode_mem, long int *J_evals)

Get the number of times the user’s relaxation Jacobian was evaluated.

Parameters:
• arkode_mem – the ARKStep memory structure

• J_evals – the number of relaxation Jacobian evaluations

Return values:
• ARK_SUCCESS – the value was successfully set

• ARK_MEM_NULLarkode_mem was NULL

• ARK_RELAX_MEM_NULL – the internal relaxation memory structure was NULL

int ARKStepGetNumRelaxFails(void *arkode_mem, long int *fails)

Get the total number of times applying relaxation failed.

The counter includes the sum of the number of nonlinear solver failures (see ARKStepGetNumRelaxSolveFails()) and the number of failures due an unacceptable relaxation value (see ARKStepSetRelaxBoundFactor()).

Parameters:
• arkode_mem – the ARKStep memory structure

• fails – the total number of failed relaxation attempts

Return values:
• ARK_SUCCESS – the value was successfully set

• ARK_MEM_NULLarkode_mem was NULL

• ARK_RELAX_MEM_NULL – the internal relaxation memory structure was NULL

int ARKStepGetNumRelaxBoundFails(void *arkode_mem, long int *fails)

Get the number of times the relaxation parameter was deemed unacceptable.

Parameters:
• arkode_mem – the ARKStep memory structure

• fails – the number of failures due to an unacceptable relaxation parameter value

Return values:
• ARK_SUCCESS – the value was successfully set

• ARK_MEM_NULLarkode_mem was NULL

• ARK_RELAX_MEM_NULL – the internal relaxation memory structure was NULL

int ARKStepGetNumRelaxSolveFails(void *arkode_mem, long int *fails)

Get the number of times the relaxation parameter nonlinear solver failed.

Parameters:
• arkode_mem – the ARKStep memory structure

• fails – the number of relaxation nonlinear solver failures

Return values:
• ARK_SUCCESS – the value was successfully set

• ARK_MEM_NULLarkode_mem was NULL

• ARK_RELAX_MEM_NULL – the internal relaxation memory structure was NULL

int ARKStepGetNumRelaxSolveIters(void *arkode_mem, long int *iters)

Get the number of relaxation parameter nonlinear solver iterations.

Parameters:
• arkode_mem – the ARKStep memory structure

• iters – the number of relaxation nonlinear solver iterations

Return values:
• ARK_SUCCESS – the value was successfully set

• ARK_MEM_NULLarkode_mem was NULL

• ARK_RELAX_MEM_NULL – the internal relaxation memory structure was NULL