# 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 assimulo, pyDAS, DAEtools
Julia ind-$$1$$ DifferentialEquations.jl BDF calls SUNDIALS IDA