Solutions

Abstract atomistic or single-step solution.

Print error for AtomicSolutions of equations not implemented, yet.

Create AtomicSolution for DAE.

Create AtomicSolution for ODE.

Create AtomicSolution for partitioned DAE.

Create AtomicSolution for partitioned ODE.

Create AtomicSolution for PSDE.

Create AtomicSolution for SDE.

Create AtomicSolution for DAE.

Create AtomicSolution for ODE.

Create AtomicSolution for partitioned DAE.

Create AtomicSolution for partitioned ODE.

Create AtomicSolution for PSDE.

Create AtomicSolution for SDE.

Atomic solution for an DAE.

Fields

• t: time of current time step
• t̅: time of previous time step
• q: current solution of q
• q̅: previous solution of q
• q̃: compensated summation error of q
• λ: current solution of λ
• λ̅: previous solution of λ
• v: vector field of q
• v̅: vector field of q̅
• u: projective vector field of q
• u̅: projective vector field of q̅

Atomic solution for an ODE.

Fields

• t: time of current time step
• t̅: time of previous time step
• q: current solution of q
• q̅: previous solution of q
• q̃: compensated summation error of q
• v: vector field of q
• v̅: vector field of q̅

Atomic solution for an PDAE.

Fields

• t: time of current time step
• t̅: time of previous time step
• q: current solution of q
• q̅: previous solution of q
• q̃: compensated summation error of q
• p: current solution of p
• p̅: previous solution of p
• p̃: compensated summation error of p
• λ: current solution of λ
• λ̅: previous solution of λ
• v: vector field of q
• v̅: vector field of q̅
• f: vector field of p
• f̅: vector field of p̅
• u: projective vector field of q
• u̅: projective vector field of q̅
• g: projective vector field of p
• g̅: projective vector field of p̅

Atomic solution for an PODE.

Fields

• t: time of current time step
• t̅: time of previous time step
• q: current solution of q
• q̅: previous solution of q
• q̃: compensated summation error of q
• p: current solution of p
• p̅: previous solution of p
• p̃: compensated summation error of p
• v: vector field of q
• v̅: vector field of q̅
• f: vector field of p
• f̅: vector field of p̅

Atomic solution for an SDE.

Fields

• t: time of current time step
• t̅: time of previous time step
• q: current solution of q
• q̅: previous solution of q
• q̃: compensated summation error of q
• p: current solution of p
• p̅: previous solution of p
• p̃: compensated summation error of p
• ΔW: Wiener process driving the stochastic process q
• ΔZ: Wiener process driving the stochastic process q
• K: integer parameter defining the truncation of the increments of the Wiener process (for strong solutions)

Atomic solution for an SDE.

Fields

• t: time of current time step
• t̅: time of previous time step
• q: current solution of q
• q̅: previous solution of q
• q̃: compensated summation error of q
• ΔW: Wiener process driving the stochastic process q
• ΔZ: Wiener process driving the stochastic process q
• K: integer parameter defining the truncation of the increments of the Wiener process (for strong solutions)

Create solution for DAE.

Create solution for ODE.

Create solution for partitioned DAE.

Create solution for partitioned ODE.

Print error for solutions of equations not implemented, yet.

Create solution for SDE.

Create solution for PSDE.

SolutionDAE: Solution of a differential algebraic equation

Contains all fields necessary to store the solution of an DAE.

Fields

• nd: dimension of the dynamical variable $q$
• nm: dimension of the constraint submanifold
• nt: number of time steps to store
• ni: number of initial conditions
• t: time steps
• q: solution q[nd, nt+1, ni] with q[:,0,:] the initial conditions
• λ: Lagrange multiplier λ[nd, nt+1, ni]
• ntime: number of time steps to compute
• nsave: store every nsave'th time step (default: 1)
• nwrite: save data to disk after every nwrite'th time step (default: ntime)
• counter: counter for copied solution entries
• woffset: counter for file offset
• h5: HDF5 file for storage

SolutionODE: Solution of an ordinary differential equation

Contains all fields necessary to store the solution of an ODE.

Fields

• nd: dimension of the dynamical variable $q$
• nt: number of time steps to store
• ni: number of initial conditions
• t: time steps
• q: solution q[nd, nt+1, ni] with q[:,0,:] the initial conditions
• ntime: number of time steps to compute
• nsave: store every nsave'th time step (default: 1)
• nwrite: save data to disk after every nwrite'th time step (default: ntime)
• counter: counter for copied solution entries
• woffset: counter for file offset
• h5: HDF5 file for storage

SolutionPDAE: Solution of a partitioned differential algebraic equation

Contains all fields necessary to store the solution of an PDAE.

Fields

• nd: dimension of the dynamical variable $q$
• nm: dimension of the constraint submanifold
• nt: number of time steps to store
• ni: number of initial conditions
• t: time steps
• q: solution q[nd, nt+1, ni] with q[:,0,:] the initial conditions
• p: solution p[nd, nt+1, ni] with p[:,0,:] the initial conditions
• λ: Lagrange multiplier λ[nd, nt+1, ni]
• ntime: number of time steps to compute
• nsave: store every nsave'th time step (default: 1)
• nwrite: save data to disk after every nwrite'th time step (default: ntime)
• counter: counter for copied solution entries
• woffset: counter for file offset
• h5: HDF5 file for storage

SolutionPODE: Solution of a partitioned ordinary differential equation

Contains all fields necessary to store the solution of an PODE.

Fields

• nd: dimension of the dynamical variable $q$
• nt: number of time steps to store
• ni: number of initial conditions
• t: time steps
• q: solution q[nd, nt+1, ni] with q[:,0,:] the initial conditions
• p: solution p[nd, nt+1, ni] with p[:,0,:] the initial conditions
• ntime: number of time steps to compute
• nsave: store every nsave'th time step (default: 1)
• nwrite: save data to disk after every nwrite'th time step (default: ntime)
• counter: counter for copied solution entries
• woffset: counter for file offset
• h5: HDF5 file for storage

SolutionPSDE: Solution of a partitioned stochastic differential equation

Contains all fields necessary to store the solution of a PSDE or SPSDE

Fields

• conv: type of the solution: :strong or :weak
• nd: dimension of the dynamical variable $q$
• nm: dimension of the Wiener process
• nt: number of time steps to store
• ns: number of sample paths
• ni: number of initial conditions
• t: time steps
• q: solution q[nd, nt+1, ns, ni] with q[:,0,:,:] the initial conditions
• p: solution p[nd, nt+1, ns, ni] with p[:,0,:,:] the initial conditions
• W: Wiener process driving the stochastic processes q and p
• K: integer parameter defining the truncation of the increments of the Wiener process (for strong solutions),

•   A = √(2 K Δt |log Δt|) due to Milstein & Tretyakov; if K=0 no truncation

• ntime: number of time steps to compute
• nsave: save every nsave'th time step

SolutionSDE: Solution of a stochastic differential equation

Contains all fields necessary to store the solution of an SDE.

Fields

• conv: type of the solution: :strong or :weak
• nd: dimension of the dynamical variable $q$
• nm: dimension of the Wiener process
• nt: number of time steps to store
• ns: number of sample paths
• t: time steps
• q: solution q[nd, nt+1, ns] with q[:,0,:] the initial conditions
• W: Wiener process driving the stochastic process q
• K: integer parameter defining the truncation of the increments of the Wiener process (for strong solutions),

•   A = √(2 K Δt |log Δt|) due to Milstein & Tretyakov; if K=0 no truncation

• ntime: number of time steps to compute
• nsave: save every nsave'th time step

Type for holding the increments of a Wiener process

Fields:

• nd: dimension of the Wiener process
• nt: number of increments in the DataSeries
• ns: number of sample paths of the Wiener process
• Δt: time increment of the TimeSeries
• ΔW: variable storing the increments of the Wiener process over Δt, or the discrete random variable \hat{I}
• ΔZ: variable holding the time integral of the Wiener process \int_{tk}^{tk+1} (W(t)-W(tk))dt, or the discrete random variable \tilde{I}

Parameters:

• dType: type of the elements of the increments of the Wiener process
• tType: type of the time steps
• N: the number of dimensions of the arrays holding data in ΔW and ΔZ
• CONV: mode of convergence: :strong or :weak

Append solution to HDF5 file. soffset - start writing the solution q at the position soffset+2 woffset - start writing the increments ΔW, ΔZ at the position woffset+1

Append solution to HDF5 file.

Append solution to HDF5 file.

Creates HDF5 file, writes solution to file, and closes file.

writetohdf5: Wrapper for saving Solution to HDF5 file.

Create parallel solution for DAE.

Create parallel solution for ODE.

Create parallel solution for partitioned DAE.

Create parallel solution for partitioned ODE.

Print error for parallel solutions of equations not implemented, yet.

Create parallel solution for SDE.

Create parallel solution for PSDE.

Copy solution from atomistic solution to solution object.

createHDF5: Creates or opens HDF5 file.

Creates HDF5 file and initialises datasets for solution object.

Creates HDF5 file and initialises datasets for deterministic solution object.

Creates HDF5 file and initialises datasets for stochastic solution object.

save_attributes: Saves attributes of Deterministic Solutions to HDF5 file.

save_attributes: Saves common attributes of Solution to HDF5 file.

save_attributes: Saves attributes of Stochastic Solutions to HDF5 file.