qmat.solvers.generic.integrators
Specialized PhiSolver classes implementing various time-integrators.
Classes
\(\phi\)-based solver doing Forward Euler update between time nodes. |
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\(\phi\)-based solver doing Backward Euler update between time nodes. |
Module Contents
- class ForwardEuler(diffOp: DiffOp, nodes, tEnd=1, nSteps=1, t0=0)[source]
\(\phi\)-based solver doing Forward Euler update between time nodes.
It uses the following definition :
\[\phi(u_0, u_1, ..., u_{m}, u_{m+1}) = \Delta\tau_{m+1} f(u_m, t_m) + ... + \Delta\tau_1 f(u_0, t_0),\]where \(\Delta\tau_{m} = t_{m+1} - t_{m}\). In particular, since it does not depends on the node solution \(u_{m+1}\) (explicit scheme), its phiSolve method is replaced by an explicit evaluation of evalPhi.
- Parameters:
diffOp (DiffOp) – Differential operator for the ODE.
nodes (1D array-like) – The time nodes \(\tau_1, ..., \tau_M\).
tEnd (float, optional) – Final simulation time. The default is 1.
nSteps (int, optional) – Number of simulation time-steps. The default is 1.
t0 (float, optional) – Initial simulation time. The default is 0.
- evalPhi(uVals, fEvals, out, t0=0)[source]
Evaluate the \(\phi\) operator on time-node up to \(u_{m+1}\).
Considering \(u_0, u_1, \dots, u_{m+1}\), if evaluates :
\[\phi(u_0, u_1, ..., u_{m}, u_{m+1}),\]and store its value into the output vector out. It also takes the node evaluation \(f(u_0,t_0),f(u_1,\tau_1),...,f(u_{m},\tau_{m})\) as arguments, in order to avoid any additional \(f(u,t)\) evaluations.
- Parameters:
uVals (list[np.ndarray] of size \(m+2\)) – The \(m+1\) time-node solutions + the initial solution \(u_0\).
fEvals (list[np.ndarray] of size \(m+1\) or \(m+1\)) – The \(f(u,t)\) evaluations at each time nodes (+ initial solution), up to time-node \(m\). It can eventually contain a pre-computed \(f_{m+1}\) to spare one \(f(u,t)\) evaluation.
out (np.ndarray) – Array used to store the evaluation.
t0 (float, optional) – Initial step time. The default is 0.
- phiSolve(uPrev, fEvals, out, rhs=0, t0=0)[source]
Solve the node update at given time-node \(\tau_{m+1}\).
Considering \(m+1\) previous known node solutions \(u_0, u_1, ..., u_{m}\), it solves the following system :
\[u -\phi(u_0, u_1, ..., u_{m}, u) = rhs,\]where the value given in out is used as initial guess and to store the computed solution. It also takes as argument the \(f\) evaluations \(f_0, f_1, ..., f_{m}\) to avoid supplementar re-computing those.
- Parameters:
uPrev (list[np.ndarray] of size \(m+1\)) – The previous node solutions \(u_0, u_1, ..., u_{m}\).
fEvals (list[np.ndarray] of size \(m+1\)) – Evaluations of previous node solutions \(f_0, f_1, ..., f_{m}\).
out (np.ndarray) – Array with the initial guess, used to store the final solution.
rhs (np.ndarray or float, optional) – Right hand side used to solve the equation above. The default is 0.
t0 (float, optional) – Initial step size. The default is 0.
- class BackwardEuler(diffOp: DiffOp, nodes, tEnd=1, nSteps=1, t0=0)[source]
\(\phi\)-based solver doing Backward Euler update between time nodes.
It uses the following definition :
\[\phi(u_0, u_1, ..., u_{m}, u_{m+1}) = \Delta\tau_{m+1} f(u_{m+1}, t_{m+1}) + ... + \Delta\tau_1 f(u_1, t_1),\]where \(\Delta\tau_{m} = t_{m+1} - t_{m}\). In particular, its phiSolve method is rewritten to depend directly on the fSolve method of the differential operator to avoid unecessary (re-)evaluations of \(f(u,t)\).
- Parameters:
diffOp (DiffOp) – Differential operator for the ODE.
nodes (1D array-like) – The time nodes \(\tau_1, ..., \tau_M\).
tEnd (float, optional) – Final simulation time. The default is 1.
nSteps (int, optional) – Number of simulation time-steps. The default is 1.
t0 (float, optional) – Initial simulation time. The default is 0.
- evalPhi(uVals, fEvals, out, t0=0)[source]
Evaluate the \(\phi\) operator on time-node up to \(u_{m+1}\).
Considering \(u_0, u_1, \dots, u_{m+1}\), if evaluates :
\[\phi(u_0, u_1, ..., u_{m}, u_{m+1}),\]and store its value into the output vector out. It also takes the node evaluation \(f(u_0,t_0),f(u_1,\tau_1),...,f(u_{m},\tau_{m})\) as arguments, in order to avoid any additional \(f(u,t)\) evaluations.
- Parameters:
uVals (list[np.ndarray] of size \(m+2\)) – The \(m+1\) time-node solutions + the initial solution \(u_0\).
fEvals (list[np.ndarray] of size \(m+1\) or \(m+1\)) – The \(f(u,t)\) evaluations at each time nodes (+ initial solution), up to time-node \(m\). It can eventually contain a pre-computed \(f_{m+1}\) to spare one \(f(u,t)\) evaluation.
out (np.ndarray) – Array used to store the evaluation.
t0 (float, optional) – Initial step time. The default is 0.
- phiSolve(uPrev, fEvals, out, rhs=0, t0=0)[source]
Solve the node update at given time-node \(\tau_{m+1}\).
Considering \(m+1\) previous known node solutions \(u_0, u_1, ..., u_{m}\), it solves the following system :
\[u -\phi(u_0, u_1, ..., u_{m}, u) = rhs,\]where the value given in out is used as initial guess and to store the computed solution. It also takes as argument the \(f\) evaluations \(f_0, f_1, ..., f_{m}\) to avoid supplementar re-computing those.
- Parameters:
uPrev (list[np.ndarray] of size \(m+1\)) – The previous node solutions \(u_0, u_1, ..., u_{m}\).
fEvals (list[np.ndarray] of size \(m+1\)) – Evaluations of previous node solutions \(f_0, f_1, ..., f_{m}\).
out (np.ndarray) – Array with the initial guess, used to store the final solution.
rhs (np.ndarray or float, optional) – Right hand side used to solve the equation above. The default is 0.
t0 (float, optional) – Initial step size. The default is 0.