foodie_integrator_adams_bashforth_moulton Module

• Uses:

• foodie_error_codes
• foodie_integrand_object
• foodie_integrator_adams_bashforth
• foodie_integrator_adams_moulton
• foodie_integrator_multistep_object
• foodie_integrator_object
• penf

FOODIE integrator: provide a predictor-corrector class of Adams-Bashforth-Moutlon multi-step schemes, from 1st to 4rd order accurate.

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 Adams-Bashforth-Moulton class scheme implemented is:

predictor

$$ U^{n+N_s^p} = U^{n+N_s^p-1} +\Delta t \left[ \sum_{s=1}^{N_s^p}{ b_s^p \cdot R(t^{n+s-1}, U^{n+s-1}) } \right] $$

corrector

$$ U^{n+N_s^c} = U^{n+N_s^c-1} +\Delta t \left[ \sum_{s=0}^{N_s^c}{ b_s^c \cdot R(t^{n+s}, U^{n+s}) } \right] $$

where \(N_s^p\) is the number of previous steps considered for the predictor and \(N_s^c\) is the number of previous steps considered for the corrector.

The coefficients \(b_s^{p,c}\) define the actual scheme, that is selected accordingly to the number of steps used. The predictor and corrector schemes should have the same formal order of accuracy, thus \(N_s^p=N_s^c+1\) should hold.

Currently, the following schemes are available:

P=AB(1)-C=AM(0) step, Explicit/Implicit Farward/Backward Euler, 1st order

This scheme is TVD and reverts to Explicit/Implicit Farward/Backward Euler, it being 1st order. The b coefficient is: $$b^p = \left[b_1\right] = \left[1\right]$$ $$b^c = \left[b_0\right] = \left[1\right]$$ The scheme is: $$ U^{n+1,p} = U^{n} + \Delta t R(t^{n},U^{n}) $$ $$ U^{n+1,c} = U^{n} + \Delta t R(t^{n+1},U^{n+1,p}) $$

P=AB(2)-C=AM(1) steps

This scheme is 2nd order. The b coefficients are: $$b^p = \left[ {\begin{array}{*{20}{c}} b_1 & b_2 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} -\frac{1}{2} & \frac{3}{2} \end{array}} \right]$$ $$b^c = \left[ {\begin{array}{*{20}{c}} b_0 & b_1 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} \frac{1}{2} & \frac{1}{2} \end{array}} \right]$$ The scheme is: $$ U^{n+2,p} = U^{n+1} +\Delta t \left[ \frac{3}{2} R(t^{n+1}, U^{n+1})-\frac{1}{2} R(t^{n}, U^{n}) \right] $$ $$ U^{n+2,c} = U^{n+1} +\Delta t \left[ \frac{1}{2} R(t^{n+2,p}, U^{n+2})+\frac{1}{2} R(t^{n+1}, U^{n+1}) \right] $$

P=AB(3)-C=AM(2) steps

This scheme is 3rd order. The b coefficients are: $$b^p = \left[ {\begin{array}{*{20}{c}} b_1 & b_2 & b_3 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} \frac{5}{12} & -\frac{4}{3} & \frac{23}{12} \end{array}} \right]$$ $$b^c = \left[ {\begin{array}{*{20}{c}} b_0 & b_1 & b_2 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} -\frac{1}{12} & \frac{2}{3} & \frac{5}{12} \end{array}} \right]$$ The scheme is: $$ U^{n+3,p} = U^{n+2} +\Delta t \left[ \frac{23}{12}R(t^{n+2}, U^{n+2}) - \frac{4}{3}R(t^{n+1}, U^{n+1}) +\frac{5}{12} R(t^{n}, U^{n}) \right] $$ $$ U^{n+3,c} = U^{n+2} +\Delta t \left[ \frac{5}{12}R(t^{n+3}, U^{n+3,p}) + \frac{2}{3}R(t^{n+2}, U^{n+2}) -\frac{1}{12} R(t^{n+1}, U^{n+1}) \right] $$

P=AB(4)-C=AM(3) steps

This scheme is 4th order. The b coefficients are: $$b^p = \left[ {\begin{array}{*{20}{c}} b_1 & b_2 & b_3 & b_4 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} -\frac{9}{24} & \frac{37}{24} & -\frac{59}{24} & \frac{55}{24} \end{array}} \right]$$ $$b^c = \left[ {\begin{array}{*{20}{c}} b_0 & b_1 & b_2 & b_3 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} \frac{1}{24} & -\frac{5}{24} & \frac{19}{24} & \frac{9}{24} \end{array}} \right]$$ The scheme is: $$ U^{n+4,p} = U^{n+3} +\Delta t \left[ \frac{55}{24}R(t^{n+3}, U^{n+3}) - \frac{59}{24}R(t^{n+2}, U^{n+2}) +\frac{37}{24} R(t^{n+1}, U^{n+1}) - \frac{9}{24} R(t^{n}, U^{n}) \right] $$ $$ U^{n+4,c} = U^{n+3} +\Delta t \left[ \frac{9}{24}R(t^{n+4}, U^{n+4,p}) + \frac{19}{24}R(t^{n+3}, U^{n+3}) -\frac{5}{24} R(t^{n+2}, U^{n+2}) + \frac{1}{24} R(t^{n+1}, U^{n+1}) \right] $$

Bibliography

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