9 Numerical Analysis and Software Overview

9.1 Theory: RKMs and BDF for DAEs

Table 9.1: Overview of convergence results of RKM/BDF schemes for DAEs

	DAEs
unstructured, linear	\(E(t)\dot x = A(t)x + f(t)\)
semi-linear	\(E(t)\dot x = f(t,x)\)
unstructured	\(F(t,\dot x, x)=0\)
unstructured, strangeness-free/index-1	\(\begin{cases}\hat F_1(t,\dot x, x)=0 \\ \hat F_2(t,x)=0 \end{cases}\)
semi-explicit, strangeness-free/index-1	\(\begin{cases}\dot x= f(t, x, y) \\ 0=g(t,x,y) \end{cases}\)
semi-explicit, index-2	\(\begin{cases}\dot x= f(t, x, y) \\ 0=g(t,y) \end{cases}\)

Table 9.2: Overview of convergence results of BDF/RKM schemes for DAEs of various index and, possibly, semi-explicit structure. Here, we equate index-1 and strangeness-free. A \(\cdot\) indicates that this case is included in a result for a more general case left or above in the table.

	RKM						BDF
	unstructured			semi-explicit			unstructured			semi-explicit
Problem / Index	\(*\)	\(2\)	\(1\)	\(*\)	\(2\)	\(1\)	\(*\)	\(2\)	\(1\)	\(*\)	\(2\)	\(1\)
nonlinear			c		g,i	b			f		h	e
linear TV			\(\cdot\)		\(\cdot\)	\(\cdot\)			\(\cdot\)		\(\cdot\)	\(\cdot\)
linear CC	a	\(\cdot\)	\(\cdot\)	\(\cdot\)	\(\cdot\)	\(\cdot\)	d	\(\cdot\)	\(\cdot\)	\(\cdot\)	\(\cdot\)	\(\cdot\)

	Description	Reference
a	RKM, linear constant coefficients	KM Thm. 5.12
b	RKM, nonlinear, strangeness-free/index-1, semi-explicit	KM Thm 5.16 / HW Thm. VI.1.1
c	RKM, nonlinear, strangeness-free	KM Thm. 5.18
d	BDF, linear constant coefficients	KM Thm. 5.24
e	BDF(\(\subset\) MSM), nonlinear, strangeness-free/index-1, semi-explicit	KM Thm. 5.26 (\(\subset\) HW Thm. VI.2.1)
f	BDF, nonlinear, strangeness-free/index-1	KM Thm. 5.27
g	RKM, nonlinear, index-2, semi-explicit	HW Ch. VII.4
h	BDF, nonlinear, index-2, semi-explicit	HW Thm. VII.3.5
i	half-explicit RKM, nonlinear, index-2, semi-explicit	HW Thm. VII.6.2
HW	Ernst Hairer, Gerhard Wanner (1996)	Solving ordinary differential equations. II: Stiff and differential-algebraic problems
KM	Peter Kunkel, Volker Mehrmann (2006)	Differential-Algebraic Equations. Analysis and Numerical Solution

9.2 Solvers

As can be seen from the table above, generally usable discretization methods for unstructured DAEs are only there for index-1 problems. However, the solvers GELDA/GENDA include an automated reduction to the strangeness-free form so that they apply for any index; see Lecture Chapter 4++.

9.2.1 Multi purpose

	DAEs	Methods	h/p	Language	Remark	Avail
GELDA	l-\(\mu\)-\(*\)	BDF/RKM	\(*\)/\(*\)	F-77		\(*\)/\(\cdot\)
GENDA	n-\(\mu\)-\(*\)	BDF	\(*\)/\(*\)	F-77		\(\phantom{*}\)/\(\cdot\)
DASSL	n-\(\nu\)-\(1\)	BDF	\(*\)/\(*\)	F-77	base for Sundials IDA – the base of many DAE solvers	\(*\)/\(\phantom{\cdot}\)
LIMEX	sl-\(\nu\)-\(1\)	x-SE-Eul	\(*\)/\(*\)	F-77		\(\phantom{*}\)/\(\phantom{\cdot}\)
RADAU	sl-\(\nu\)-\(1\)	RKM	\(*\)/\(*\)	F-77		\(*\)/\(\phantom{\cdot}\)

Notes:

	Explanation
DAEs	l-linear, sl-semilinear, nl-nonlinear
	classification: \(\mu\)-strangeness index, \(\nu\)-differentiation index
	\(*\)-includes index reduction
h/p	time step control / order control
availability	code for download / licence provided(\(*\)) or other statement(\(\cdot\))
methods	x-SE-Eul: extrapolation based on semiexplicit Euler

9.2.2 Application specific

Furthermore, there are solvers for particularly structured DAEs.

DAEs	Resources
Navier-Stokes (nl-se-\(2\))	See, e.g., Sec. 4.3 of our preprint on definitions of different schemes
Multi-Body (nl-se-\(3\))	See, e.g., the code on Hairer’s homepage

9.3 Software

Many software suits actually wrap SUNDIALS IDA.

	DAEs	Routines	Method	Remark
Matlab	ind-\(1\)	ode15{i,s}	BDF
Python	—			no built-in functionality, DASSL/IDA wrapped in the modules scikit-odes, assimulo, pyDAS, DAEtools
Julia	ind-\(1\)	DifferentialEquations.jl	BDF	calls SUNDIALS IDA