An Hermite-Obreschkoff method for stiff high-index DAE

Abstract

We have developed an implicit Hermite-Obreschkoff method for the numerical solution of stiff high-index differential-algebraic equations (DAEs). On each integration step, this method requires the computation of Taylor coefficients and their gradients to construct and solve a nonlinear system for the numerical solution, which is then projected to satisfy the constraints of the problem. We derive this system, show how to compute its Jacobian through automatic differentiation, and present the ingredients of our method, such as predicting an initial guess for Newton’s method, error estimation, and stepsize and order control. We report numerical results on stiff DAEs illustrating the accuracy and performance of our method, and in particular, its ability to take large steps on stiff problems.

A dprsd: equations of motion

Denote the cartesian coordinates of $\mathsf {\MakeUppercase {A}}$ and $\mathsf {\MakeUppercase {B}}$ by $\mathsf {\MakeUppercase {A}} = (\mathsf {\MakeUppercase {A}}_{x}, \mathsf {\MakeUppercase {A}}_{y})$ and $\mathsf {\MakeUppercase {B}} = (\mathsf {\MakeUppercase {B}}_{x}, \mathsf {\MakeUppercase {B}}_y)$. The kinetic and potential energies are

$$\begin{aligned} T&= \tfrac{1}{6} {m_1} |\dot{\mathsf {\MakeUppercase {A}}}|^{2} + \tfrac{1}{6} {m_1}{m_2}(|\dot{\mathsf {\MakeUppercase {A}}}|^{2}+\dot{\mathsf {\MakeUppercase {A}}}\cdot \dot{\mathsf {\MakeUppercase {B}}}+ |\dot{\mathsf {\MakeUppercase {B}}}|^{2}), \end{aligned}$$

(A.1)

$$\begin{aligned} V&= \tfrac{1}{2} m_{1}g \mathsf {\MakeUppercase {A}}_{y} + \tfrac{1}{2} m_2g(\mathsf {\MakeUppercase {A}}_{y}+\mathsf {\MakeUppercase {B}}_{y}) + k_{1}(\theta -\alpha )^{2} +k_2(\pi - \theta +\phi - \beta )^{2}, \end{aligned}$$

(A.2)

respectively ($|\cdot |$ is vector length and $\cdot$ is dot product), and the Lagrangian is

$$\begin{aligned} \mathscr {L}&= T-V. \end{aligned}$$

(A.3)

We use Rayleigh’s dissipation function

$$\begin{aligned} \mathscr {R}&=\tfrac{1}{2} c_{1} {\dot{\theta }^{2}} + \tfrac{1}{2} c_{2}(\dot{\phi }-\dot{\theta })^{2}. \end{aligned}$$

(A.4)

ODE formulation. With generalized coordinates $\textbf{q} = (\theta , \phi )$, $\mathsf {\MakeUppercase {A}}= \ell _1\,(\cos \theta , \sin \theta )$ and $\mathsf {\MakeUppercase {B}}= \mathsf {\MakeUppercase {A}}+\ell _2\, (\cos \phi , \sin \phi ).$ Using (A.1–A.4) in

$$\begin{aligned} \qquad 0 = \frac{\text {d}}{\text {d}t} \frac{\partial \mathscr {L}}{\partial \dot{\textbf{q}}} -\frac{\partial \mathscr {L}}{\partial \textbf{q}} + \frac{\partial \mathscr {R}}{\partial \dot{\textbf{q}}}, \end{aligned}$$

we derive the second-order ODE

$$\begin{aligned} \begin{aligned} 0&= \begin{bmatrix}\left( \tfrac{1}{3}{m_1}+ m_2\right) \ell _1^{2} &{} \tfrac{1}{2} {m_2}\ell _1 \ell _2 \cos (\theta -\phi ) \\ \frac{1}{2}{m_2}\ell _1 \ell _2 \cos (\theta -\phi ) &{} \frac{1}{3}{m_2} \ell _2^{2} \end{bmatrix} \begin{bmatrix}\ddot{\theta }\\ \ddot{\phi }\end{bmatrix} +\tfrac{1}{2}{m_2}\ell _1\ell _2 \sin (\theta -\phi ) \begin{bmatrix}{\dot{\phi }^{2}} \\ -{\dot{\theta }^{2}} \end{bmatrix}\\&\qquad + \begin{bmatrix} \left( \tfrac{1}{2}m_1 +m_2 \right) g\ell _1 \cos \theta + k_1(\theta -\alpha )-k_2 (\pi - \theta +\phi -\beta )\\ \tfrac{1}{2} m_2g\ell _2\cos \phi + k_2(\pi - \theta +\phi -\beta ) \end{bmatrix} \\&\qquad + \begin{bmatrix}c_{1}\dot{\theta }- c_2(\dot{\phi }-\dot{\theta }) \\ c_2(\dot{\phi }-\dot{\theta }) \end{bmatrix} . \end{aligned} \end{aligned}$$

(A.5)

DAE formulation. We chose as generalized coordinates $\textbf{q} = (\mathsf {\MakeUppercase {A}}_x, \mathsf {\MakeUppercase {A}}_y, \mathsf {\MakeUppercase {B}}_{x} ,\mathsf {\MakeUppercase {B}}_{y})$. Denote $\mathsf {\MakeUppercase {C}}= \mathsf {\MakeUppercase {B}}-\mathsf {\MakeUppercase {A}}$. We have the index-3, second-order DAE system

$$\begin{aligned} \begin{aligned} 0&= \frac{\text {d}}{\text {d}t} \frac{\partial \mathscr {L}}{\partial \dot{\textbf{q}}_{i}} -\frac{\partial \mathscr {L}}{\partial \textbf{q}_{i}} + \sum _{k=1}^{2} \lambda _{k}\frac{\partial C_{k}}{\partial {\textbf{q}}_{i}}+\frac{\partial \mathscr {R}}{\partial \dot{\textbf{q}_{i}}}, \quad i = 1,\ldots , 4, \\ 0&= C_{1} = |\mathsf {\MakeUppercase {A}}|^{2} -\ell ^{2}, \\ 0&= C_{2} = |\mathsf {\MakeUppercase {C}}|^{2} - \ell _2^{2},\\ 0&= \dot{\theta } -\frac{1}{\ell _1^{2}}(\mathsf {\MakeUppercase {A}}_x\cdot \dot{\mathsf {\MakeUppercase {A}}}_y - \mathsf {\MakeUppercase {A}}_y\cdot \dot{\mathsf {\MakeUppercase {A}}}_x), \\ 0&= \dot{\phi } - \frac{1}{\ell _2^{2}} (\mathsf {\MakeUppercase {C}}_x\cdot \dot{\mathsf {\MakeUppercase {C}}}_y - \mathsf {\MakeUppercase {C}}_y\cdot \dot{\mathsf {\MakeUppercase {C}}}_x) \end{aligned} \end{aligned}$$

(A.6)

in state variables $(\mathsf {\MakeUppercase {A}}_x, \mathsf {\MakeUppercase {A}}_y, \mathsf {\MakeUppercase {B}}_{x} ,\mathsf {\MakeUppercase {B}}_{y}, \lambda _{1}, \lambda _{2}, \theta , \phi )$. Here $\lambda _{1}$ and $\lambda _{2}$ are Lagrange multipliers.