foodie_integrator_runge_kutta_lssp Module

• Uses:

• foodie_error_codes
• foodie_integrand_object
• foodie_integrator_multistage_object
• foodie_integrator_object
• penf

FOODIE integrator: provide an explicit class of Linear SSP Runge-Kutta schemes, from 1st to s-th order accurate.

The integrators provided have the Total Variation Diminishing (TVD) property or the Strong Stability Preserving (SSP) one. The schemes are explicit.

Considering the following ODE system:

$$ U_t = R(t,U) $$

where \(U_t = \frac{dU}{dt}\), U is the vector of state variables being a function of the time-like independent variable t, R is the (vectorial) residual function, the class of schemes implemented are written in the following 2 forms:

Ns stages, Ns-1 order

$$ U^0 = U^n $$ $$ U^k = U^{k-1} + \frac{1}{2} \Delta t R(U^{k-1}) \quad k=1,..,Ns $$ $$ U^{Ns} = U^{Ns} + \frac{1}{2} \Delta t R(U^{Ns}) $$ $$ U^{n+1} = \sum_{k=1}^{Ns}\alpha_k U^k $$

where Ns is the number of stages used and \(U^k\) is the \(k^{th}\) stage. The minimum number of stages is 2. The coefficients \(\alpha\) are computed by means of the following recursive algorith:

Computation of coefficients

• allocate coefficients array, allocate(alpha(1:Ns));
• initialize the first 2 elements with the coefficients of the two stages first order methods, alpha(1)=0, alpha(2)=1;
• for i in [3,Ns] do:
• alpha(i) = (2 / i) * alpha(i-1)
• for j in [i-1,2,-1] do:
  • alpha(j) = (2 / (j-1)) * alpha(j-1)
• for j in [2,i] do:
  • alpha(1) = 1 - alpha(j)

Ns stages, Ns order

$$ U^0 = U^n $$ $$ U^k = U^{k-1} + \Delta t R(U^{k-1}) \quad k=1,..,Ns $$ $$ U^{Ns} = U^{Ns} +\Delta t R(U^{Ns}) $$ $$ U^{n+1} = \sum_{k=1}^{Ns}\alpha_k U^k $$

where Ns is the number of stages used and \(U^k\) is the \(k^{th}\) stage. The minimum number of stages is 1. The coefficients \(\alpha\) are computed by means of the following recursive algorith:

Computation of coefficients

• allocate coefficients array, allocate(alpha(1:Ns));
• initialize the first element with the coefficient of the one stage first order method, alpha(1)=1;
• for i in [2,Ns] do:
• alpha(i) = 1 / i!
• for j in [i-1,2,-1] do:
  • alpha(j) = (1 / (j-1)) * alpha(j-1)
• for j in [2,i] do:
  • alpha(1) = 1 - alpha(j)

Bibliography

[2] Strong Stability Preserving Runge-Kutta and Multistep Time Discretizations, S. Gottlieb, D. Ketcheson, C.W. Shu, 2011, 978-981-4289-26-9, doi:10.1142/7498, World Scientific Publishing Co. Pte. Ltd.

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