Equations

GeometricIntegrators.Equations.DAE — Type.

DAE: Differential Algebraic Equation

Defines a differential algebraic initial value problem

\[\begin{align*} \dot{q} (t) &= v(t, q(t)) + u(t, q(t), \lambda(t)) , & q(t_{0}) &= q_{0} , \\ 0 &= \phi (t, q(t), \lambda(t)) , & \lambda(t_{0}) &= \lambda_{0} , \end{align*}\]

with vector field $v$, projection $u$, algebraic constraint $\phi=0$, initial conditions $q_{0}$ and $\lambda_{0}$, the dynamical variable $q$ taking values in $\mathbb{R}^{m}$ and the algebraic variable $\lambda$ taking values in $\mathbb{R}^{n}$.

Fields

d: dimension of dynamical variable $q$ and the vector field $v$
m: dimension of algebraic variable $\lambda$ and the constraint $\phi$
n: number of initial conditions
v: function computing the vector field
u: function computing the projection
ϕ: algebraic constraint
t₀: initial time
q₀: initial condition for dynamical variable $q$
λ₀: initial condition for algebraic variable $\lambda$

The function v, providing the vector field, takes three arguments, v(t, q, v), the functions u and ϕ, providing the projection and the algebraic constraint take four arguments, u(t, q, λ, u) and ϕ(t, q, λ, ϕ), where t is the current time, q and λ are the current solution vectors, and v, u and ϕ are the vectors which hold the result of evaluating the vector field $v$, the projection $u$ and the algebraic constraint $\phi$ on t, q and λ.

Example

    function v(t, q, v)
        v[1] = q[1]
        v[2] = q[2]
    end

    function u(t, q, λ, u)
        u[1] = +λ[1]
        u[2] = -λ[1]
    end

    function ϕ(t, q, λ, ϕ)
        ϕ[1] = q[2] - q[1]
    end

    t₀ = 0.
    q₀ = [1., 1.]
    λ₀ = [0.]

    dae = DAE(v, u, ϕ, t₀, q₀, λ₀)

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GeometricIntegrators.Equations.HDAE — Type.

HDAE: Hamiltonian Differential Algebraic Equation

Defines a Hamiltonian differential algebraic initial value problem, that is a canonical Hamiltonian system of equations subject to Dirac constraints,

\[\begin{align*} \dot{q} (t) &= v_1(t, q(t), p(t)) + v_2(t, q(t), p(t), \lambda(t)) + v_3(t, q(t), p(t), \lambda(t), \gamma(t)) , & q(t_{0}) &= q_{0} , \\ \dot{p} (t) &= f_1(t, q(t), p(t)) + f_2(t, q(t), p(t), \lambda(t)) + f_3(t, q(t), p(t), \lambda(t), \gamma(t)) , & p(t_{0}) &= p_{0} , \\ 0 &= \phi (t, q(t), p(t)) , \\ 0 &= \psi (t, q(t), p(t), \lambda(t)) , \end{align*}\]

with vector fields $v_i$ and $f_i$ for $i = 1 ... 3$, primary constraint $\phi(q,p)=0$ and secondary constraint $\psi(q,p,\lambda)=0$, initial conditions $(q_{0}, p_{0})$, the dynamical variables $(q,p)$ taking values in $\mathbb{R}^{d} \times \mathbb{R}^{d}$ and the algebraic variables $(\lambda, \gamma)$ taking values in $\mathbb{R}^{n} \times \mathbb{R}^{d}$.

Fields

d: dimension of dynamical variables $q$ and $p$ as well as the vector fields $v$ and $f$
m: dimension of algebraic variables $\lambda$ and $\gamma$ and the constraints $\phi$ and $\psi$
n: number of initial conditions
v: tuple of functions computing the vector fields $v_i$, $i = 1 ... 3$
f: tuple of functions computing the vector fields $f_i$, $i = 1 ... 3$
ϕ: primary constraints
ψ: secondary constraints
t₀: initial time
q₀: initial condition for dynamical variable $q$
p₀: initial condition for dynamical variable $p$

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GeometricIntegrators.Equations.IDAE — Type.

IDAE: Implicit Differential Algebraic Equation

Defines a partitioned differential algebraic initial value problem

\[\begin{align*} \dot{q} (t) &= v(t) + u(t, q(t), p(t), \lambda(t)) , & q(t_{0}) &= q_{0} , \\ \dot{p} (t) &= f(t, q(t), v(t)) + r(t, q(t), p(t), \lambda(t)) , & p(t_{0}) &= p_{0} , \\ p(t) &= p(t, q(t), v(t)) , && \ 0 &= \phi (t, q(t), p(t), \lambda(t)) , & \lambda(t_{0}) &= \lambda_{0} , \end{align*}\]

with vector field $f$, the momentum defined by $p$, projection $u$ and $r$, algebraic constraint $\phi=0$, conditions $(q_{0}, p_{0})$ and $\lambda_{0}$, the dynamical variables $(q,p)$ taking values in $\mathbb{R}^{d} \times \mathbb{R}^{d}$ and the algebraic variable $\lambda$ taking values in $\mathbb{R}^{n}$.

Fields

d: dimension of dynamical variables $q$ and $p$ as well as the vector fields $f$ and $p$
m: dimension of algebraic variable $\lambda$ and the constraint $\phi$
n: number of initial conditions
f: function computing the vector field $f$
p: function computing $p$
u: function computing the projection
g: function computing the projection
ϕ: algebraic constraint
t₀: initial time
q₀: initial condition for dynamical variable $q$
p₀: initial condition for dynamical variable $p$
λ₀: initial condition for algebraic variable $\lambda$

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GeometricIntegrators.Equations.IODE — Type.

IODE: Implicit Ordinary Differential Equation

Defines an implicit initial value problem

\[\begin{align*} \dot{q} (t) &= v(t) , & q(t_{0}) &= q_{0} , \\ \dot{p} (t) &= f(t, q(t), v(t)) , & p(t_{0}) &= p_{0} , \\ p(t) &= α(t, q(t), v(t)) \end{align*}\]

with vector field $f$, the momentum defined by $p$, initial conditions $(q_{0}, p_{0})$ and the solution $(q,p)$ taking values in $\mathbb{R}^{d} \times \mathbb{R}^{d}$. This is a special case of a differential algebraic equation with dynamical variables $(q,p)$ and algebraic variable $v$.

Fields

d: dimension of dynamical variables $q$ and $p$ as well as the vector fields $f$ and $p$
α: function determining the momentum
f: function computing the vector field
g: function determining the projection, given by ∇α(q)λ
v: function computing an initial guess for the velocity field (optional)
t₀: initial time (optional)
q₀: initial condition for q
p₀: initial condition for p

The functions α and f must have the interface

    function α(t, q, v, p)
        p[1] = ...
        p[2] = ...
        ...
    end

and

    function f(t, q, v, f)
        f[1] = ...
        f[2] = ...
        ...
    end

where t is the current time, q is the current solution vector, v is the current velocity and f and p are the vectors which hold the result of evaluating the functions $f$ and $α$ on t, q and v. The funtions g and v are specified by

    function g(t, q, λ, g)
        g[1] = ...
        g[2] = ...
        ...
    end

and

    function v(t, q, p, v)
        v[1] = ...
        v[2] = ...
        ...
    end

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GeometricIntegrators.Equations.ODE — Type.

ODE: Ordinary Differential Equation

Defines an initial value problem

\[\dot{q} (t) = v(t, q(t)) , \qquad q(t_{0}) = q_{0} ,\]

with vector field $v$, initial condition $q_{0}$ and the solution $q$ taking values in $\mathbb{R}^{d}$.

Fields

d: dimension of dynamical variable $q$ and the vector field $v$
n: number of initial conditions
v: function computing the vector field
t₀: initial time
q₀: initial condition

The function v providing the vector field must have the interface

    function v(t, q, v)
        v[1] = ...
        v[2] = ...
        ...
    end

where t is the current time, q is the current solution vector, and v is the vector which holds the result of evaluating the vector field $v$ on t and q.

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GeometricIntegrators.Equations.PDAE — Type.

PDAE: Partitioned Differential Algebraic Equation

Defines a partitioned differential algebraic initial value problem

\[\begin{align*} \dot{q} (t) &= v(t, q(t), p(t)) + u(t, q(t), p(t), \lambda(t)) , & q(t_{0}) &= q_{0} , \\ \dot{p} (t) &= f(t, q(t), p(t)) + r(t, q(t), p(t), \lambda(t)) , & p(t_{0}) &= p_{0} , \\ 0 &= \phi (t, q(t), p(t), \lambda(t)) , & \lambda(t_{0}) &= \lambda_{0} , \end{align*}\]

with vector fields $v$ and $f$, projection $u$ and $r$, algebraic constraint $\phi=0$, conditions $(q_{0}, p_{0})$ and $\lambda_{0}$, the dynamical variables $(q,p)$ taking values in $\mathbb{R}^{d} \times \mathbb{R}^{d}$ and the algebraic variable $\lambda$ taking values in $\mathbb{R}^{n}$.

Fields

d: dimension of dynamical variables $q$ and $p$ as well as the vector fields $f$ and $p$
m: dimension of algebraic variable $\lambda$ and the constraint $\phi$
n: number of initial conditions
v: function computing the vector field $v$
f: function computing the vector field $f$
u: function computing the projection
g: function computing the projection
ϕ: algebraic constraint
t₀: initial time
q₀: initial condition for dynamical variable $q$
p₀: initial condition for dynamical variable $p$
λ₀: initial condition for algebraic variable $\lambda$

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GeometricIntegrators.Equations.PODE — Type.

IODE: Partitioned Ordinary Differential Equation

Defines a partitioned initial value problem

\[\begin{align*} \dot{q} (t) &= v(t, q(t), p(t)) , & q(t_{0}) &= q_{0} , \\ \dot{p} (t) &= f(t, q(t), p(t)) , & p(t_{0}) &= p_{0} , \end{align*}\]

with vector fields $v$ and $f$, initial conditions $(q_{0}, p_{0})$ and the solution $(q,p)$ taking values in $\mathbb{R}^{d} \times \mathbb{R}^{d}$.

Fields

d: dimension of dynamical variables $q$ and $p$ as well as the vector fields $v$ and $f$
v: function computing the vector field $v$
f: function computing the vector field $f$
t₀: initial time
q₀: initial condition for q
p₀: initial condition for p

The functions v and f must have the interface

    function v(t, q, p, v)
        v[1] = ...
        v[2] = ...
        ...
    end

and

    function f(t, q, p, f)
        f[1] = ...
        f[2] = ...
        ...
    end

where t is the current time, q and p are the current solution vectors and v and f are the vectors which hold the result of evaluating the vector fields $v$ and $f$ on t, q and p.

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GeometricIntegrators.Equations.PSDE — Type.

PSDE: Stratonovich Partitioned Stochastic Differential Equation

Defines a partitioned stochastic differential initial value problem

\[\begin{align*} \dq (t) &= v(t, q(t)) \, dt + B(t, q(t)) \circ dW , & q(t_{0}) &= q_{0} , \dp (t) &= f(t, q(t)) \, dt + G(t, q(t)) \circ dW , & p(t_{0}) &= p_{0} \end{align*}\]

with the drift vector fields $v$ and $f$, diffusion matrices $B$ and $G$, initial conditions $q_{0}$ and $p_{0}$, the dynamical variables $(q,p)$ taking values in $\mathbb{R}^{d} \times \mathbb{R}^{d}$, and the m-dimensional Wiener process W

Fields

d: dimension of dynamical variable $q$ and the vector field $v$
m: dimension of the Wiener process
n: number of initial conditions
ns: number of sample paths
v: function computing the drift vector field for the position variable q
f: function computing the drift vector field for the momentum variable p
B: function computing the d x m diffusion matrix for the position variable q
G: function computing the d x m diffusion matrix for the momentum variable p
t₀: initial time
q₀: initial condition for dynamical variable $q$ (may be a random variable itself)
p₀: initial condition for dynamical variable $p$ (may be a random variable itself)

The functions v, f, 'B' and G, providing the drift vector fields and diffusion matrices, take four arguments, v(t, q, p, v), f(t, q, p, f), B(t, q, p, B) and G(t, q, p, G), where t is the current time, (q, p) is the current solution vector, and v, f, 'B' and G are the variables which hold the result of evaluating the vector fields $v$, $f$ and the matrices $B$, $G$ on t and (q,p).

Example

    function v(λ, t, q, v)
        v[1] = λ*q[1]
        v[2] = λ*q[2]
    end

    function B(μ, t, q, B)
        B[1] = μ*q[1]
        B[2] = μ*q[2]
    end

    t₀ = 0.
    q₀ = [1., 1.]
    λ  = 2.
    μ  = 1.

    v_sde = (t, q, v) -> v(λ, t, q, v)
    B_sde = (t, q, B) -> B(μ, t, q, B)

    sde = SDE(v_sde, B_sde, t₀, q₀)

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GeometricIntegrators.Equations.SDE — Type.

SDE: Stratonovich Stochastic Differential Equation

Defines a stochastic differential initial value problem

\[\begin{align*} \dq (t) &= v(t, q(t)) \, dt + B(t, q(t)) \circ dW , & q(t_{0}) &= q_{0} , \end{align*}\]

with drift vector field $v$, diffusion matrix $B$, initial conditions $q_{0}$, the dynamical variable $q$ taking values in $\mathbb{R}^{d}$, and the m-dimensional Wiener process W

Fields

d: dimension of dynamical variable $q$ and the vector field $v$
m: dimension of the Wiener process
n: number of initial conditions
ns: number of sample paths
v: function computing the deterministic vector field
B: function computing the d x m diffusion matrix
t₀: initial time
q₀: initial condition for dynamical variable $q$ (may be a random variable itself)

The functions v and B, providing the drift vector field and diffusion matrix, v(t, q, v) and B(t, q, B; col=0), where t is the current time, q is the current solution vector, and v and B are the variables which hold the result of evaluating the vector field $v$ and the matrix $B$ on t and q (if col==0), or the column col of the matrix B (if col>0).

Example

    function v(λ, t, q, v)
        v[1] = λ*q[1]
        v[2] = λ*q[2]
    end

    function B(μ, t, q, B; col=0)
        if col==0 #whole matrix
            B[1,1] = μ*q[1]
            B[2,1] = μ*q[2]
        elseif col==1
            #just first column
        end
    end

    t₀ = 0.
    q₀ = [1., 1.]
    λ  = 2.
    μ  = 1.

    v_sde = (t, q, v) -> v(λ, t, q, v)
    B_sde = (t, q, B) -> B(μ, t, q, B)

    sde = SDE(v_sde, B_sde, t₀, q₀)

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GeometricIntegrators.Equations.SODE — Type.

SODE: Split Ordinary Differential Equation

Defines an initial value problem

\[\dot{q} (t) = v(t, q(t)) , \qquad q(t_{0}) = q_{0} ,\]

with vector field $v$, initial condition $q_{0}$ and the solution $q$ taking values in $\mathbb{R}^{d}$. Here, the vector field $v$ is given as a sum of vector fields

\[v (t) = v_1 (t) + ... + v_r (t) .\]

Fields

d: dimension of dynamical variable $q$ and the vector field $v$
v: tuple of functions computing the vector field
t₀: initial time
q₀: initial condition

The functions v_i providing the vector field must have the interface

    function v_i(t, q₀, q₁, h)
        q₁[1] = q₀[1] + ...
        q₁[2] = q₀[2] + ...
        ...
    end

where t is the current time, q₀ is the current solution vector, q₁ is the new solution vector which holds the result of computing one substep with the vector field $v_i$ on t and q₀, and h is the (sub-)timestep to compute the update for.

The fact that the function v returns the solution and not just the vector field for each substep increases the flexibility for the use of splitting methods, e.g., it allows to use another integrator for solving substeps.

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GeometricIntegrators.Equations.SPSDE — Type.

SPSDE: Stratonovich Split Partitioned Stochastic Differential Equation

Defines a partitioned stochastic differential initial value problem

\[\begin{align*} \dq (t) &= v(t, q(t)) \, dt + B(t, q(t)) \circ dW , & q(t_{0}) &= q_{0} , \dp (t) &= [ f1(t, q(t)) + f2(t, q(t)) ] \, dt + [ G1(t, q(t)) + G2(t, q(t)) ] \circ dW , & p(t_{0}) &= p_{0} \end{align*}\]

with the drift vector fields $v$ and $fi$, diffusion matrices $B$ and $Gi$, initial conditions $q_{0}$ and $p_{0}$, the dynamical variables $(q,p)$ taking values in $\mathbb{R}^{d} \times \mathbb{R}^{d}$, and the m-dimensional Wiener process W

Fields

d: dimension of dynamical variable $q$ and the vector fields $vi$
m: dimension of the Wiener process
n: number of initial conditions
ns: number of sample paths
v : function computing the drift vector field for the position variable q
f1: function computing the drift vector field for the momentum variable p
f2: function computing the drift vector field for the momentum variable p
B : function computing the d x m diffusion matrix for the position variable q
G1: function computing the d x m diffusion matrix for the momentum variable p
G2: function computing the d x m diffusion matrix for the momentum variable p
t₀: initial time
q₀: initial condition for dynamical variable $q$ (may be a random variable itself)
p₀: initial condition for dynamical variable $p$ (may be a random variable itself)

Example

    function v(λ, t, q, v)
        v[1] = λ*q[1]
        v[2] = λ*q[2]
    end

    function B(μ, t, q, B)
        B[1] = μ*q[1]
        B[2] = μ*q[2]
    end

    t₀ = 0.
    q₀ = [1., 1.]
    λ  = 2.
    μ  = 1.

    v_sde = (t, q, v) -> v(λ, t, q, v)
    B_sde = (t, q, B) -> B(μ, t, q, B)

    sde = SDE(v_sde, B_sde, t₀, q₀)

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GeometricIntegrators.Equations.VODE — Type.

VODE: Variational Ordinary Differential Equation

Defines an implicit initial value problem

\[\begin{align*} \dot{q} (t) &= v(t) , & q(t_{0}) &= q_{0} , \\ \dot{p} (t) &= f(t, q(t), v(t)) , & p(t_{0}) &= p_{0} , \\ p(t) &= α(t, q(t), v(t)) \end{align*}\]

Fields

d: dimension of dynamical variables $q$ and $p$ as well as the vector fields $f$ and $p$
α: function determining the momentum
f: function computing the vector field
g: function determining the projection, given by ∇α(q)λ
v: function computing an initial guess for the velocity field (optional)
t₀: initial time (optional)
q₀: initial condition for q
p₀: initial condition for p
λ₀: initial condition for λ

The functions α and f must have the interface

    function α(t, q, v, p)
        p[1] = ...
        p[2] = ...
        ...
    end

and

    function f(t, q, v, f)
        f[1] = ...
        f[2] = ...
        ...
    end

    function g(t, q, λ, g)
        g[1] = ...
        g[2] = ...
        ...
    end

and

    function v(t, q, p, v)
        v[1] = ...
        v[2] = ...
        ...
    end

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