Initial Value Problems Problems

Definitions

An Initial Value Problem (IVP) is defined as

(1)

\begin{eqnarray} F(t,y,\dot y) &=& 0 \\ y(t_0) &=& y_0 \end{eqnarray}

If the Jacobian $F_{\dot y}$ is nonsingular, (1) represents an Ordinary Differential Equation (ODE). In principle, one could solve for $\dot y$ as

(2)

\begin{eqnarray} \dot y &=& f(t,y) \\ y(t_0) &=& y_0 \end{eqnarray}

to obtain an ODE in so-called explicit form.

On the other hand, if $F_{\dot y}$ is singular, (1) represents a Differential Algebraic Equation (DAE). DAEs are best classified using various concepts of their index:

• index of nilpotency (for linear constant coefficient DAE): measure of the numerical difficulty in solving the DAE;
• differentiation index: “measure of the "departure"” from an ODE;
• perturbation index: measure of the sensitivity of the solutions with respect to perturbations;
• etc.

The differentiation index is defined as follows [Gear and Petzold, 1983]:

Equation (1) has differentiation index $m$ if $m$ is the minimal number of analytical differentiation

(3)

\begin{align} F(t,y,\dot y) = 0 \, , \, \frac{dF(t,y,\dot y)}{dt}=0 \, ,\, \ldots \, , \, \frac{d^mF(t,y,\dot y)}{dt^m}=0 \end{align}

such that, by algebraic manipulations, we can extract an explicit ODE $\dot y = \phi(t,y)$ (called an underlying ODE)

Note that, an ODE has therefore index-0.

Rheinboldt (1984) advocated looking at DAEs as differential equations on manifolds:

(4)

\begin{eqnarray} &&\dot y = v(t,y) \, , \quad y \in \cal{M} \\ \text{manifold:}&&{\cal M} = \{y \in R^n | g(y) = 0 \} \\ \text{tangent space:}&& T_y {\cal M} = \{ v \in R^n | g_y(y)v = 0 \} \\ \text{vector field on} {\cal M}: && v:{\cal M} \rightarrow R^n \, ; \quad \forall y \in {\cal M} \Rightarrow v(y) \in T_y {\cal M} \end{eqnarray}

DAE examples

index-1

If

(5)

\begin{align} F = \left[f(t,u,v,\dot u) \, ; \, g(t,u,v) \right]^T \end{align}

and both $f_{\dot u}$ and $g_v$ are square and nonsingular, then $F(t,y,\dot y) = 0$ has index-1. Indeed, one can solve $g=0$ for $v$, substitute in $f=0$, and solve for $\dot u$. Note that there are more general forms of index-1 DAEs.

Hessenberg index-1

(6)

\begin{eqnarray} \dot y &=& f(t, y,z) \\ 0 &=& g(y,z) \end{eqnarray}

is an index-1 DAE in Hessenberg form if $g_z$ is nonsingular.

Example
single perturbation problems (e.g. Robertson's chemical kinetics problem):

(7)

\begin{eqnarray} {\dot y}_A = -0.04y_A + 10^4 y_B y_C \,; && y_A(0) = 1 \\ {\dot y}_B = 0.04y_A - 10^4 y_B y_C - 3\cdot10^7 y_B^2 \,;&& y_B(0) = 0 \\ 1 = y_A + y_B + y_C && \end{eqnarray}

Hessenberg index-2

(8)

\begin{eqnarray} \dot y &=& f(t,y,z) \\ 0 &=& g(y) \\ \end{eqnarray}

Example
modeling of incompressible fluid flow by Navier-Stokes:

(9)

\begin{eqnarray} u_t + uu_x + vu_y + p_x - \nu (u_{xx} + u_{yy}) &=& 0 \\ v_t + uv_x + vv_y + p_y - \nu (v_{xx} + v_{yy}) &=& 0 \\ u_x + v_y &=& 0 \end{eqnarray}

with appropriate spatial discretization.

Hessenberg index-3

(10)

\begin{eqnarray} \dot x &=& f(t,x,y,z) \\ \dot y &=& g(x,y) \\ 0 &=& h(y) \end{eqnarray}

is an index-3 DAE in Hessenberg form if $h_y g_x f_z$ is nonsingular.

Example
modeling of constrained mechanical systems with holonomic constraints:

(11)

\begin{eqnarray} M(q)\dot v &=& F(t,q,v) - \Phi_q^T(q)\lambda \\ \dot q &=& v \\ 0 &=& \Phi(q) \end{eqnarray}

If the constraints $\Phi$ are linearly independen, then $\Phi_qM^{-1}\Phi_q^T$ is nonsingular.