Implicit Euler Method for Condensation¶

Header file: <libs/superdrops/impliciteuler.hpp> [source]

struct ImplicitIteration¶

Struct defining parameters and methods for iterations of the Implicit Euler Method.

This struct defines parameters and methods for performing an iterations of the implicit method. E.g. struct defines the number of iterations to try before testing convergence, the timestep of the method, it’s relative and absolute tolerances, and methods for computing the initial guess for and performing the Newton-Raphson root finding.

_Note: abbreviation NR = Newton Raphson (Method)

Public Functions

double initialguess(const double rprev) const¶

Returns appropriate initial guess (ie. a reasonable guess) for the Newton-Raphson method.

This method returns an initial guess for the Newton-Raphson method based on the given radius from the previous timestep and the current supersaturation ratio.

Guess is supposed to be a reasonable value for initial ‘ziter’ to use as first iteration of NR method in rootfinding algorithm for timestepping condensation/evaporation ODE. Here the guess criteria are as in SCALE-SDM for making initial guess for given droplet much greater than its (activation radius)^2 if the supersaturation > its activation supersaturation.

Parameters:: rprev – Radius of droplet at previous timestep.
Returns:: Initial guess for ziter.

double initialguess_shima(const double rprev) const¶

Returns appropriate initial guess (ie. a reasonable guess) for the Newton-Raphson method.

This method returns an initial guess for the Newton-Raphson method based on the given radius from the previous timestep and the current supersaturation ratio.

Guess is supposed to be a reasonable value for initial ‘ziter’ to use as first iteration of NR method in rootfinding algorithm for timestepping condensation/evaporation ODE. Here the guess criteria are adapted from SCALE-SDM. Second criteria is that initial guess >= ‘r1sqrd’, where r1 is the equilibrium radius of a given droplet when s_ratio=1.

Parameters:: rprev – Radius of droplet at previous timestep.
Returns:: Initial guess for ziter.

double newtonraphson_niterations(const double rprev, double ziter) const¶

Performs Newton-Raphson iterations with a fixed number of iterations.

This method performs Newton-Raphson iterations for a fixed number of iterations, then tests for convergence and continues with futher iterations until convergence occurs or or until the maximum number of iterations is reached.

Funciton intergrates (timesteps) condensation ODE by delt given initial guess for ziter, (which is usually radius^squared from previous timestep). Uses Newton Raphson iterative method to find new value of the radius that converges on the root of the polynomial g(ziter) within the tolerances of the ImpIter instance. After ‘niters’ iterations, convergence criteria is tested and futher iterations undertaken if polynomial root has not yet been converged upon.

Parameters:

rprev – Radius at previous timestep
ziter – The current guess for ziter.

Returns:

The updated value of ziter.

double newtonraphson_untilconverged(const unsigned int iterlimit, const double rprev, double ziter) const¶

Performs Newton-Raphson iterations until convergence.

After every iteration, convergence criteria is tested and error is raised if method does not converge within ‘iterlimit’ iterations. Otherwise returns new value for the radius (which is the radius at timestep ‘t+subdelt’. Refer to section 5.1.2 Shima et al. 2009 and section 3.3.3 of Matsushima et al. 2023 for more details.

Parameters:

iterlimit – The maxiumum number of iterations to attempt.
rprev – Radius at the previous timestep.
ziter – The current guess for ziter.

Returns:

The updated value of ziter.

Kokkos::pair<bool, double> iterate_rootfinding_algorithm(const double rprev, double ziter) const¶

Perform one iteration of the Newton-Raphson rootfinding algorithm.

This function performs one iteration of the Newton-Raphson rootfinding algorithm and returns the updated value of radius^2 alongside a boolean indicating whether the algorithm has converged.

Parameters:

rprev – Radius at the previous timestep.
ziter – The current guess for ziter.

Returns:

A pair indicating whether to continue iterating and the updated value of ziter.

double ode_gfunc(const double rprev, const double rsqrd) const¶

Returns the value of g(z) / z * delt for the ODE.

This method computes the value of g(z) / z * delt used in the root-finding Newton-Raphson method for the dr/dt condensation / evaporation ODE.

ODE is for radial growth/shrink of each superdroplet due to condensation and diffusion of water vapour according to equations from “An Introduction To Clouds….” (see note at top of file).

Note: z = ziter = radius^2.

Parameters:

rprev – Radius at the previous timestep.
rsqrd – Current radius squared.

Returns:

RHS of g(z) / z * delt evaluted at rqrd.

double ode_gfuncderivative(const double rsqrd) const¶

Returns the value of the derivative of g(z) with respect to z.

This method computes the value of dg(z)/dz * delt, where dg(z)/dz is the derivative of g(z) with respect to z=rsqr. g(z) is polynomial to find root of using Newton Raphson Method consistent as in ode_gfunc(…).

Parameters:: rsqrd – Current radius squared.
Returns:: RHS of dg(z)/dz * delt evaluted at rqrd.

inline bool isnot_converged(const double gfunciter, const double gfuncprev) const¶

Returns true if the Newton-Raphson iterations have not yet converged.

This method checks if the Newton-Raphson iterations have converged based on a standard local error test: |iteration - previous iteration| < RTOL * |iteration| + ATOL.

Parameters:

gfunciter – g(z) of current iteration
gfuncprev – g(z) of previous iteration

Returns:

boolean=true if not yet converged, false otherwise

Public Members

unsigned int niters¶: Number of NR iterations to try before testing convergence.

double subdelt¶: Timestep of implicit method (at each substep >= niter NR iterations occur).

double rtol¶: Relative tolerance for convergence of NR method.

double atol¶: Absolute tolerance for convergence of NR method.

double s_ratio¶: Supersaturation ratio.

double akoh¶: Kelvin factor in Kohler theory “a”.

double bkoh¶: Raoult factor in Kohler theory “b”.

double ffactor¶: Sum of heat and vapor diffusion factors.

class ImplicitEuler¶

Class for Implicit Euler (IE) integration of superdroplet condensational growth / evaporational shrinking ODE.

This class performs Implicit Euler integration of the superdroplet condensation / evaporation ODE using a Newton-Raphson root finding method to solve the implicit timestep equation of a stiff ODE.

Public Functions

inline ImplicitEuler(const unsigned int niters, const double delt, const double maxrtol, const double maxatol, const double subdelt)¶: Constructor for ImplicitEuler class.

double solve_condensation_matsushima(const double s_ratio, const Kokkos::pair<double, double> akoh_bkoh, const double ffactor, const double rprev) const¶

Integrates the condensation / evaporation ODE employing the Implicit Euler method as in Matsushima et al, 2023.

Forward timestep previous radius ‘rprev’ by delt using an Implicit Euler method to integrate the condensation/evaporation ODE. Implict timestepping equation defined in section 5.1.2 of Shima et al. 2009 and is root of polynomial g(z) = 0, where z = [R_i(t+delt)]^squared.

Newton Raphson iterations are used to converge towards the root of g(z) within the tolerances of an ImpIter instance. Tolerances, maxium number of iterations and sub-timestepping are adjusted based on the uniqueness criteria of the polynomial g(z). Uniqueness criteria, ucrit1 and / or ucrit2, assume that solution to g(ziter)=0 is unique and therefore Newton Raphson root finding algorithm converges quickly. This means method can be used with comparitively large tolerances and timesteps, and the maximum number of iterations is small. Refer to section 5.1.2 of Shima et al. 2009 and section 3.3.3 of Matsushima et al. 2023 for more details.

double solve_condensation(const double s_ratio, const Kokkos::pair<double, double> akoh_bkoh, const double ffactor, const double rprev) const¶

Integrates the condensation / evaporation ODE employing the Implicit Euler method as in Shima et. al, 2023 with adjustments for near-supersaturation conditions.

Newton Raphson iterations are used to converge towards the root of g(z) within the tolerances of an ImpIter instance. Tolerances, maxium number of iterations and sub-timestepping are adjusted when near to supersaturation=1 (when activation / deactivation may occur). Far from activation, solution to g(ziter)=0 is usually unique and Newton Raphson root finding algorithm converges quickly. This means method can be used with comparitively large tolerances and timesteps, and the maximum number of iterations is small.

Private Functions

double substepped_implicitmethod(const ImplicitIteration &implit, const unsigned int nsubsteps, const double rprev) const¶

Performs the implicit method with sub-stepping.

This method performs the implicit method with substepping, iterating over the substeps to compute the implicit method for each substep.

Parameters:

implit – object defining implicit iterations.
nsubsteps – number of substeps to perform.
rprev – Radius of droplet at previous timestep.

Returns:

New value for droplet radius.

Private Members

unsigned int niters¶: Suggested number of iterations for Newton Raphson method before testing for convergence.

double delt¶: Timestep of ODE solver (at each step implicit method is called).

double maxrtol¶: Adjustable relative tolerance for convergence of NR method.

double maxatol¶: Adjustable absolute tolerance for convergence of NR method.

double subdelt¶: Number of substeps to intergation when supersat close to 1 (0 = false).