6 Construction and Analysis of RKM for nonlinear DAEs

Now we consider RKM for nonlinear DAEs. We start with a DAE in semi explicit strangeness-free form and give general results on how to write down a general RKM for it and how to analyse the global error. Then, we consider general strangeness-free nonlinear DAEs and show that a certain class of RKM applies well – namely those that can be constructed by collocation with Lagrange polynomials over the Radau, Lobatto, or Gauss quadrature points.

6.1 General RKM for Semi-Explicit Strangeness-free DAEs

A semi explicit strangeness-free DAE is of the form \[\begin{align} \dot x &= f(t, x, y) \tag{6.1} \\ 0 &= g(t, x, y) \tag{6.2} \end{align}\] with the Jacobian of $g$ with respect to $y$, i.e. \[ \partial_y\otimes g(t, x(t), y(t)) =: g_y(t, x(t), y(t)), \] being invertible for all $t$ along the solution $(x,y)$.

Some observations:

this system is strangeness-free
under certain assumptions, any DAE can be brought into this form
in the linear case $E\dot z = Az +f$, with $z=(x,y)$, the assumptions basically mean that \[ E = \begin{bmatrix} I & 0 \\ 0 & 0 \end{bmatrix} \quad\text{and}\quad A = \begin{bmatrix} * & * \\ * & A_{22} \end{bmatrix}, \] with $A_{22}(t)$ invertible for all $t$.
The condition $g_y$ invertible means that, locally, one could consider \[ \dot x = f(t, x, R(t,x)), \quad\text{with $R$ such that}\quad y=R(t,x). \] However, this is not practical for numerical purposes.

The general strategy to get a suitable formulation of a time discretization of system (6.1)-(6.2) by any RKM is to consider the perturbed version \[\begin{align*} \dot x = f(t, x, y), \\ \varepsilon \dot y = g(t, x, y), \end{align*}\] which is an ODE, formulate the RKM, and then let $\varepsilon \to 0$. In the Hairer/Wanner Book, this approach is called *$\varepsilon$-embedding.

This is, consider

Table 6.1: RKM applied to semi-explicit DAEs
$x_{i+1} = x_i + h\sum_{j=1}^s\beta_j \dot X_{ij}$,	$y_{i+1} = y_i + h\sum_{j=1}^s\beta_j \dot Y_{ij}$,
$\dot X_{ij} = f(t_i+\gamma_jh, X_{ij}, Y_{ij})$,	$\varepsilon \dot Y_{ij} = g(t_i+\gamma_j h, X_{ij}, Y_{ij})$,	$j=1,2,\cdots,s, \quad \quad (*)$
$X_{ij} = x_i + h\sum_{\ell=1}^s\alpha_{j\ell}\dot X_{i\ell}$,	$\phantom{\varepsilon}Y_{ij} = y_i + h\sum_{\ell=1}^s\alpha_{j\ell}\dot Y_{i\ell}$,	$j=1,2,\cdots,s,$

i.e., the RKM applied to an ODE in the variables $(x,y)$, and replace $(*)$ by

\[ \dot X_{ij} = f(t_i+\gamma_jh, X_{ij}, Y_{ij}),\quad 0 = g(t_i+\gamma_j h, X_{ij}, Y_{ij}), \quad j=1,2,\cdots,s. \]

Theorem 6.1 (Kunkel/Mehrmann Thm. 5.16) Consider a semi-explicit, strangeness-free DAE as in (6.1)-(6.2) with a consistent initial value $(x_0, y_0)$. The time-discretization by a RKM,

with $\mathcal A$ invertible and $\rho:=1-\beta^T\mathcal A^{-1}e$,
applied as in Table 6.1 with $\varepsilon=0$,
that is convergent of order $p$ for ODEs
and fulfills the Butcher condition $C(q)$ with $q\geq p+1$

leads to an global error that behaves like \[ \|\mathfrak X(t_N) - \mathfrak X_N\| = \mathcal O(h^k), \]

where

$k=p$, if $\rho=0$,
$k=\min\{p, q+1\}$, if $-1\leq \rho < 1$
$k=\min\{p, q-1\}$, if $\rho =1$.

If $|\rho|>1$, then the RKM – applied to (6.1)–(6.2) – does not converge.

Some words on the conditions on $p$, $q$, and $\rho$:

For stiffly accurate methods, $\beta^T \mathcal A^{-1}e=1$ and, thus, $\rho=0$ $\rightarrow$ no order reduction for strangeness free or index-1 systems
For the implicit midpoint rule also known as the 1-stage Gauss method: \[ \begin{array}{c|c} \frac 12 & \frac 12 \\ \hline & 1 \end{array} \]
- the convergence order for ODEs is $p=2$
- but $1-\beta^T \mathcal A^{-1}e = 1- 1\cdot {\bigl(\frac 12\bigr)}^{-1} 1 = -1$, so that $\min\{p-1, q\} = k \leq 1$, depending on $q$.
- in fact $k=1$ as \[ C(q): \quad \sum_{\ell=1}^s\alpha_{j\ell}\gamma_\ell^{\bar k-1}=\frac{1}{\bar k} \gamma_j^{\bar k}, \quad {\bar k}=1,\cdots,q, \quad j=1,\cdots,s \] in the present case of $s=1$, $\alpha_{11}=\gamma_1=\frac 12$ is fulfilled for ${\bar k}=1: \quad \frac 12 = \frac 12$
- it is not relevant here, but for ${\bar k}=2:\quad \frac 12 \cdot \frac 12 \neq \frac 12 \cdot \frac 14$

6.2 Collocation RKM for Implicit Strangeness-free DAEs

The general form of a strangeness-free DAE is given as \[\begin{align} \hat F_1(t,x,\dot x) &= 0 \tag{6.3}\\ \hat F_2(t,x) &= 0 \tag{6.4} \end{align}\] where the strangeness-free or index-1 assumption is encoded in the existence of implicit functions $\mathcal L$, $\mathcal R$ such that, with $x=(x_1,x_2)$, the implicit DAE (6.3)–(6.4) is equivalent to the semi-explicit DAE \[\begin{align*} \dot x_1 &= \mathcal L(t,x_1,x_2) \\ 0 &= \mathcal R(t,x_1) -x_2 \end{align*}\]

In what follows we show that a collocation approach coincides with certain RKM discretizations so that the convergence analysis of the RKM can be done via approximation theory.

Regression (Collocation): – If one looks for a function $x\colon [0,1] \to \mathbb R^{}$ that fulfills $F(x(t))=0$ for all $t\in[0,1]$, one may interpolate $x$ by, say, a polynomial $x_p(t) = \sum_{\ell=0}^kx_\ell t^\ell$ and determine the $k+1$ coefficients $x_\ell$ via the solution of the system of (nonlinear) equations $F(x_p(t_\ell))=0$, $\ell=0,1,\dotsc,k$, where the $t_\ell\in[0,1]$ are the $k+1$ collocation points.

Concretely, we parametrize $s$ collocation points via \[\begin{equation} 0 = \gamma_0 < \gamma_1 <\gamma_2< \dotsc < \gamma_s=1 \tag{6.5} \end{equation}\] and define two sets of Lagrange polynomials \[ L_\ell(\xi) = \prod_{j=0,j\neq \ell}^s \frac{\xi-\gamma_j}{\gamma_\ell-\gamma_j} \quad\text{and}\quad \tilde L_\ell(\xi) = \prod_{m=1,m\neq \ell}^s \frac{\xi-\gamma_m}{\gamma_\ell-\gamma_m}, \] with $\ell\in\{0,1,\dotsc,s\}$.

Let $\mathbb P_k$ be the space of polynomials of degree $\leq k-1$. We define the collocation polynomial $x_\pi \in \mathbb P_{s+1}$ via \[\begin{equation} x_\pi (t) = \sum_{\ell=0}^s X_{i\ell}L_\ell\bigl(\frac{t-t_i}{h}\bigr) \tag{6.6} \end{equation}\] designed to compute the stage values $X_{i\ell}$, where $X_{i0}=x_i$ is already given.

The stage derivatives are then defined as \[\begin{equation} \dot X_{ij} = \dot x_\pi(t_i+\gamma_jh) = \frac 1h \sum_{\ell=0}^sX_{i\ell}\dot L_\ell(\gamma_j). \tag{6.7} \end{equation}\]

To obtain $x_{i+1}=x_\pi(t_{i+1})=X_{is}$, we require the polynomial to satisfy the DAE (6.3)–(6.4) at the collocation points $t_{ij}=t_i+\gamma_jh$, that is

\[\begin{equation} \hat F_1(t_i+\gamma_jh,X_{ij},\dot X_{ij}) = 0, \quad \hat F_2(t_i+\gamma_jh,X_{ij}) = 0, \quad j=1,\dotsc,s. \phantom{F_1} \tag{6.8} \end{equation}\]

Now we show that this collocation defines a RKM discretization of (6.3)–(6.4).

Since $\tilde L_\ell \in \mathbb P_s$ for $\ell=1,\ldots,s$ it holds that \[ P_\ell(\sigma):=\int_0^\sigma \tilde L_\ell (\xi)d\xi \in \mathbb P_{s+1} \] that is, by Lagrange interpolation, it can be written as \[ P_\ell(\sigma) = \sum_{j=0}^s P_\ell(\gamma_j)L_j(\sigma). \]

If we differentiate $P_\ell$, we get \[ \dot P_\ell(\sigma) = \sum_{j=0}^s P_\ell(\gamma_j)\dot L_j(\sigma) = \sum_{j=0}^s \int_0^{\gamma_j} \tilde L_\ell (\xi)d\xi \dot L_j(\sigma)=: \sum_{j=0}^s \alpha_{j\ell} \dot L_j(\sigma) \] where define simply define \[ \alpha_{j\ell} = \int_0^{\gamma_j} \tilde L_\ell (\xi)d\xi. \] Note that $\alpha_{0\ell}=0$ such that summation in $\dot P_\ell(\sigma)$ starts at $j=1$ instead of $j=0$. Moreover, by definition of $P_\ell$ (and the fundamental theorem of calculus), it holds that \[ \dot P_\ell(\sigma) = \tilde L_\ell(\sigma), \] which gives that $\dot P_\ell(\gamma_m) = \delta_{\ell m}$ that is \[ \dot P_\ell(\gamma_m) = \sum_{j=1}^s\alpha_{j\ell}\dot L_j(\gamma_m) = \begin{cases} 1, &\quad \text{if }\ell =m \\ 0, &\quad \text{otherwise} \end{cases} \] for $\ell, m=1,\dotsc,s$.

Accordingly, if we define $\mathcal A := \bigl[\alpha_{j\ell}\bigr]_{j,\ell=1,\dotsc,s} \in \mathbb R^{s,s}$ and \[ V:=\bigl[v_{mj}\bigr]_{m,j=1,\dotsc,s} = \bigl[ \dot L_j(\gamma_m) \bigr]_{m,j=1,\dotsc,s} \in \mathbb R^{s,s} , \] it follows that $V=\mathcal A^{-1}$.

Moreover, since, \[ \sum_{j=0}^s L_j(\sigma) \equiv 1, \quad\text{so that }\quad\sum_{j=0}^s \dot L_j(\sigma) \equiv 0, \] we have that \[ \sum_{j=0}^s \dot L_j(\gamma_m) =0= \sum_{j=0}^s v_{mj} \] and, thus, \[ v_{m0} = -\sum_{j=1}^s \dot L_j(\gamma_m) = -e_m^TVe. \] With these relations we rewrite (6.7) as \[ h\dot X_{im} = \sum_{\ell=0}^sX_{i\ell}\dot L_\ell(\gamma_m) = v_{m0}x_i + \sum_{\ell=1}^sv_{m\ell}X_{i\ell}. \] and $h\sum_{m=1}^s\alpha_{\ell m} \dot X_{im}$ as \[\begin{align} h\sum_{m=1}^s \alpha_{\ell m}\dot X_{im} &= \sum_{m=1}^s \alpha_{\ell m}v_{m0}x_i + \sum_{j,m=1}^s \alpha_{\ell m}v_{mj}X_{ij} \notag \\ &= -e_\ell^T \mathcal AV e x_i + \sum_{j=1}^se_\ell^T \mathcal AVe_jX_{ij} \tag{6.9}\\ &= -x_i + X_{i\ell}, \notag \end{align}\] which, together with (6.8), indeed defines a RKM.

Some remarks:

the preceding derivation shows that the collocation (6.6) and (6.8) is equivalent to the RKM scheme (6.9) and (6.8)
convergence of these schemes applied to (6.3)–(6.4) is proven in Kunkel/Mehrmann Theorem 5.17
with fixing $\gamma_s=1$, the obtained RKM is stiffly accurate
the remaining $s-1$ $\gamma$s can be chosen to get optimal convergence rates $\rightarrow$ RadauIIa schemes
if also $\gamma_s$ is chosen optimal in terms of convergence, the Gauss schemes are obtained

\(x_{i+1} = x_i + h\sum_{j=1}^s\beta_j \dot X_{ij}\),	\(y_{i+1} = y_i + h\sum_{j=1}^s\beta_j \dot Y_{ij}\),
\(\dot X_{ij} = f(t_i+\gamma_jh, X_{ij}, Y_{ij})\),	\(\varepsilon \dot Y_{ij} = g(t_i+\gamma_j h, X_{ij}, Y_{ij})\),	\(j=1,2,\cdots,s, \quad \quad (*)\)
\(X_{ij} = x_i + h\sum_{\ell=1}^s\alpha_{j\ell}\dot X_{i\ell}\),	\(\phantom{\varepsilon}Y_{ij} = y_i + h\sum_{\ell=1}^s\alpha_{j\ell}\dot Y_{i\ell}\),	\(j=1,2,\cdots,s,\)