Consider a system of ordinary differential equations,

$M\frac{d\mathbf{x}}{dt}+\sigma (t)(A\mathbf{x}(t)-\mathbf{f}(t))=0,\phantom{\rule{1em}{0ex}}t>0,\mathbf{x}(0)={\mathbf{x}}_{0},$

(4)

where $\mathbf{x},\mathbf{f}\in {\mathrm{\Re}}^{n}$, $\sigma (t)\ge {\sigma}_{0}>0$, *M*, *A* are $n\times n$ matrices, where *M* is assumed to be spd and the symmetric part of *A* is positive semidefinite. In the practical applications that we consider, *M* corresponds to a mass matrix and *A* to a second-order diffusion or diffusion-convection matrix. Hence, *n* is large. Under reasonable assumptions on the source function **f**, such a system is stable for all *t* and its solution approaches a finite function, independent on the initial value ${\mathbf{x}}_{0}$, as $t\to \mathrm{\infty}$.

Such stability results hold for more general problems, such as for a nonlinear parabolic problem,

$\frac{\partial u}{\partial t}+F(t,u)=0,\phantom{\rule{1em}{0ex}}\text{where}F(t,u)=-\mathrm{\nabla}\cdot (a(t,u,\mathrm{\nabla}u)\mathrm{\nabla}u)-f(t,u),x\in \mathrm{\Omega},t0,$

(5)

where $f:(0,\mathrm{\infty})\times V\to {V}^{\prime}$ and *V* is a reflexive Banach space.

For proper functions

$a(\cdot )$ and

$f(\cdot )$, then

*F* is monotone,

*i.e.* $(F(t,u)-F(t,v),u-v)\ge \rho (t){\parallel u-v\parallel}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}u,v\in V,t>0.$

(6)

Here,

$\rho :(0,\mathrm{\infty})\to R$,

$\rho (t)\ge 0$ and

$(\cdot ,\cdot )$,

$\parallel \cdot \parallel $ denote the scalar product, and the corresponding norm in

${L}^{2}(\mathrm{\Omega})$, respectively. In this case, one can easily derive the bound

$\frac{1}{2}\frac{d}{dt}\left({\parallel u-v\parallel}^{2}\right)=-(F(t,u)-F(t,v),u-v)\le -\rho (t){\parallel u-v\parallel}^{2},$

where

*u*,

*v* are solution of (5) corresponding to different initial values. Consequently making use of the Gronwall lemma, we obtain

$\parallel u(t)-v(t)\parallel \le exp(-{\int}_{0}^{t}\rho (s)\phantom{\rule{0.2em}{0ex}}ds)\parallel u(0)-v(0)\parallel \le \parallel u(0)-v(0)\parallel ,\phantom{\rule{1em}{0ex}}t>0.$

Hence, (5) is stable in this case.

If

*F* is strongly monotone (or dissipative),

*i.e.* (6) is valid with

$\rho (t)\ge {\varrho}_{0}>0$, then

$\parallel u(t)-v(t)\parallel \le exp(-t{\rho}_{0})\parallel u(0)-v(0)\parallel \to 0,\phantom{\rule{1em}{0ex}}t\to \mathrm{\infty},$

*i.e.* (5) is asymptotically stable. In particular, the above holds for the test problem considered in Section 6.

For large eigenvalues of

${M}^{-1}A$, such a system is stiff and can have fast decreasing and possibly oscillating components. This amounts to that the eigenvalues have large real part and possibly also large imaginary parts. To handle this, one needs stable numerical time-integration methods that do not contain corresponding increasing components. For

$\sigma (t)=1$, in (4), this amounts to proper approximations of the matrix exponential function

$exp(tE)$,

$E={M}^{-1}A$, by a rational function,

${R}_{m}(tE)={Q}_{m}{(tE)}^{-1}{P}_{m}(tE),$

where

$\parallel {R}_{m}(tE)\parallel \le 1,\phantom{\rule{1em}{0ex}}t>0,\text{for}Re\{{\lambda}_{E}\}0,$

and ${\lambda}_{E}$ denotes eigenvalues by *E*. Furthermore, to cope with problems where $arg({\lambda}_{E})\le \alpha <\frac{\pi}{2}$, but arbitrarily close to $\pi /2$, one needs *A*-stable methods; see *e.g.* [3, 7, 8]. To get stability for all times and time steps, one requires ${lim}_{|\lambda |\to \mathrm{\infty}}|{R}_{m}(\lambda )|\le c<1$ where preferably $c=0$. Such methods are called *L*-stable (Lambert) and stiffly *A*-stable [3], respectively.

An important class of methods which are stiffly *A*-stable is a particular class of the implicit Runga-Kutta methods; see [1, 3, 5]. Such methods correspond to rational polynomial approximations of the matrix exponential function with denominator having a higher degree than the nominator. Examples of such methods are based on Radau quadrature where the quadrature points are zeros of ${\tilde{P}}_{m}(\xi )-{\tilde{P}}_{m-1}(\xi )$, where $\{{\tilde{P}}_{k}\}$ are the Legendre polynomials, orthogonal on the interval $(0,1)$, see *e.g.* [1] and references therein. Note that $\xi =1$ is a root for all $m\ge 1$. The case $m=1$ is identical to the implicit Euler method.

Following [5], we consider here the next simplest case, where $m=2$, for the numerical solution of (4) over a time interval $[t,t+\tau ]$.

In this case, the quadrature points (for a unit interval) are

${\xi}_{1}=1/3$,

${\xi}_{2}=1$ and the numerical solution

${\mathbf{x}}_{1}$,

${\mathbf{x}}_{2}$, at

$t+\tau /3$ and

$t+\tau $ satisfies

$\left[\begin{array}{cc}M+5{\sigma}_{1}\tilde{A}& -{\sigma}_{2}\tilde{A}\\ 9{\sigma}_{1}\tilde{A}& M+3{\sigma}_{2}\tilde{A}\end{array}\right]\left[\begin{array}{c}{\mathbf{x}}_{1}\\ {\mathbf{x}}_{2}\end{array}\right]=\left[\begin{array}{c}M{\mathbf{x}}_{0}+\frac{\tau}{12}(5{\mathbf{f}}_{1}-{\mathbf{f}}_{2})\\ M{\mathbf{x}}_{0}+\frac{\tau}{4}(3{\mathbf{f}}_{1}+{\mathbf{f}}_{2})\end{array}\right],$

(7)

where ${\mathbf{x}}_{0}$ is the solution at time *t*, ${\sigma}_{1}=\sigma (t+\tau /3)$, ${\sigma}_{2}=\sigma (t+\tau )$, ${\mathbf{f}}_{1}=\mathbf{f}(t+\tau /3)$, ${\mathbf{f}}_{2}=\mathbf{f}(t+\tau )$, and $\tilde{A}=\frac{\tau}{12}A$. The global discretization error of the ${\mathbf{x}}_{2}$-component for this method is $O({\tau}^{3})$, *i.e.* it is a third-order method and it is stiffly *A*-stable even for arbitrary strong variations of the coefficient $\sigma (t)$. This can be compared with the trapezoidal or implicit midpoint methods which are only second order accurate and not stiffly stable.

The system in (7) can be solved

*via* its Schur complement. Thereby, to avoid an inner system with matrix

$M+5{\sigma}_{1}\tilde{A}$, we derive a modified form of the Schur complement system, that involves only an inner system with matrix

${M}^{-1}$. To this end, but only for the derivation of the method, we scale first the system with the block diagonal matrix

$\left[\begin{array}{cc}{M}^{-1}& 0\\ 0& {M}^{-1}\end{array}\right]$ to get

$\left[\begin{array}{cc}I+5{\sigma}_{1}G& -{\sigma}_{2}G\\ 9{\sigma}_{1}G& I+3{\sigma}_{2}G\end{array}\right]\left[\begin{array}{c}{\mathbf{x}}_{1}\\ {\mathbf{x}}_{2}\end{array}\right]=\left[\begin{array}{c}{\mathbf{x}}_{0}+\frac{\tau}{12}(5{\tilde{\mathbf{f}}}_{1}-{\tilde{\mathbf{f}}}_{2})\\ {\mathbf{x}}_{0}+\frac{\tau}{4}(3{\tilde{\mathbf{f}}}_{1}+{\tilde{\mathbf{f}}}_{2})\end{array}\right],$

where

$G=\frac{\tau}{12}{M}^{-1}A$ and

${\tilde{\mathbf{f}}}_{i}={M}^{-1}{\mathbf{f}}_{i}$,

$i=1,2$. The Schur complement system for

${\mathbf{x}}_{2}$ is multiplied with

$(I+5{\sigma}_{1}G)$. Using commutativity, we get then

$\begin{array}{c}[(I+5{\sigma}_{1}G)(I+3{\sigma}_{2}G)+9{\sigma}_{1}{\sigma}_{2}{G}^{2}]{\mathbf{x}}_{2}\hfill \\ \phantom{\rule{1em}{0ex}}=(I+5{\sigma}_{1}G)[{\mathbf{x}}_{0}+\frac{\tau}{4}(3{\tilde{\mathbf{f}}}_{1}+{\tilde{\mathbf{f}}}_{2})]-9{\sigma}_{1}G[{\mathbf{x}}_{0}+\frac{\tau}{12}(5{\tilde{\mathbf{f}}}_{1}-{\tilde{\mathbf{f}}}_{2})]\hfill \end{array}$

or

$\begin{array}{c}[I+(5{\sigma}_{1}+3{\sigma}_{2})G+24{\sigma}_{1}{\sigma}_{2}{G}^{2}]{\mathbf{x}}_{2}\hfill \\ \phantom{\rule{1em}{0ex}}=(I-4{\sigma}_{1}G){\mathbf{x}}_{0}+\frac{\tau}{4}(3{\tilde{\mathbf{f}}}_{1}+{\tilde{\mathbf{f}}}_{2})+2\tau {\sigma}_{1}G{\tilde{\mathbf{f}}}_{2}.\hfill \end{array}$

Hence,

$B{\mathbf{x}}_{2}=(M-\frac{\tau}{3}{\sigma}_{1}A){\mathbf{x}}_{0}+\frac{\tau}{4}M(3{\tilde{\mathbf{f}}}_{1}+{\tilde{\mathbf{f}}}_{2})+\frac{1}{6}{\tau}^{2}{\sigma}_{1}A{\tilde{\mathbf{f}}}_{2},$

where

$B=M+\frac{\tau}{12}(5{\sigma}_{1}+3{\sigma}_{2})A+\frac{{\tau}^{2}}{6}{\sigma}_{1}{\sigma}_{2}A{M}^{-1}A.$

(8)

For higher order Radau quadrature methods, the corresponding matrix polynomial in ${M}^{-1}B$ is a *m* th order polynomial. By the fundamental theorem of algebra, one can factorize it in factors of at most second degree. They can be solved in a sequential order. Alternatively, using a method referred to in Remark 3.1, the solution components can be computed concurrently.

Each second-order factor can be preconditioned by the method in Section 2. The ability to factorize ${Q}_{m}(tE)$ in second-order factors and solve the arising systems as such two-by-two block matrix systems means that one only has to solve first-order systems. This is of importance if for instance *M* and *A* are large sparse bandmatrices, since then one avoids increasing bandwidths in matrix products and one can solve systems of linear combinations of *M* and *A* more efficiently than for higher order polynomial combinations. Furthermore, this enables one to keep matrices on element by element form (see, *e.g.* [9]) and it is in general not necessary to store the matrices *M* and *A*. The arising inner system can be solved by some inner iteration method.

The problem with a direct factorization in first order factors is that complex matrix factors appear. This occurs for the matrix in (8) for a ratio of

$\frac{{\sigma}_{1}}{{\sigma}_{2}}$ in the interval

$3\frac{11-\sqrt{96}}{25}<\frac{{\sigma}_{1}}{{\sigma}_{2}}<3\frac{11+\sqrt{96}}{25}.$

(9)

Therefore, it is more efficient to keep the second order factors and instead solve the corresponding systems by preconditioned iterations. Thereby, the preconditioner involves only first order factors. As shown in Section 2, a very efficient preconditioner for the matrix

*B* in (8) is

$C={C}_{\alpha}=(M+\alpha \tau A){M}^{-1}(M+\sigma \tau A),$

(10)

where $\alpha >0$ is a parameter. As already shown in [5], for the above particular application it holds.

**Proposition 3.1** *Let* *B*,

*C* *be as defined in* (8)

*and* (10)

*and assume that* *M* *is spd and* *A* *is spsd*.

*Then letting* $\alpha =max\{\sqrt{{\sigma}_{1}{\sigma}_{2}/6},(5{\sigma}_{1}+3{\sigma}_{2})/24\}$

*it holds*
$\kappa \left({C}^{-1}B\right)\le \underset{i=1,2}{max}{\delta}_{i}^{-1},$

*where*
$\begin{array}{c}1\ge {\delta}_{1}=(5{\sigma}_{1}+3{\sigma}_{2})/24\alpha \ge \sqrt{10}/4,\hfill \\ 1\ge {\delta}_{2}={\sigma}_{1}{\sigma}_{2}/6{\alpha}^{2}.\hfill \end{array}$

*If* $0.144\le \frac{{\sigma}_{1}}{{\sigma}_{2}}\le 2.496$, *then* ${\delta}_{2}=1$ *and* ${\delta}_{1}\ge \sqrt{\frac{5}{8}}$.

*The spectral condition number is then bounded by*
$\kappa \left({C}^{-1}B\right)\le \sqrt{\frac{8}{5}}\approx 1.265.$

*If* ${\sigma}_{1}={\sigma}_{2}$,

*then* $\kappa \left({C}^{-1}B\right)\le \sqrt{\frac{3}{2}}\approx 1.225.$

*Proof* Let

$(\mathbf{u},\mathbf{v})$ be the

${\ell}_{2}$ product of

$\mathbf{u},\mathbf{v}\in {\mathrm{\Re}}^{n}$. We have

$(C\mathbf{x},\mathbf{x})-(B\mathbf{x},\mathbf{x})=2\sigma \tau (1-{\delta}_{1})(A\mathbf{x},\mathbf{x})+{\alpha}^{2}{\tau}^{2}(1-{\delta}_{2})(A{M}^{-1}A\mathbf{x},\mathbf{x})\phantom{\rule{1em}{0ex}}\mathrm{\forall}\mathbf{x}\in {\mathrm{\Re}}^{n}.$

It follows that

$(B\mathbf{x},\mathbf{x})\le (C\mathbf{x},\mathbf{x}).$

By the arithmetic-geometric means inequality, we have

$\delta \ge \frac{1}{2}\sqrt{15{\sigma}_{1}{\sigma}_{2}}/\alpha \ge \frac{1}{\sqrt{2}}\sqrt{90}=\frac{\sqrt{10}}{4}.$

(11)

a computation shows that

${\sigma}_{1}{\sigma}_{2}/6\ge {\left(\frac{5{\sigma}_{1}+3{\sigma}_{2}}{24}\right)}^{2}$

for

$0.144\lesssim \xi \lesssim 2.496$, where

$\xi ={\sigma}_{1}/{\sigma}_{2}$. Further, a computation shows that

${\delta}_{1}\ge \sqrt{\frac{5}{8}}$, which is in accordance with the lower bound in (11). Since

$(C\mathbf{x},\mathbf{x})\ge 2\alpha \tau (A\mathbf{x},\mathbf{x})+{\alpha}^{2}{\tau}^{2}(A{M}^{-1}A\mathbf{x},\mathbf{x}),$

it follows that

$1-\frac{(B\mathbf{x},\mathbf{x})}{(C\mathbf{x},\mathbf{x})}\ge 1-{\delta}_{1}$

or

$\frac{(B\mathbf{x},\mathbf{x})}{(C\mathbf{x},\mathbf{x})}\le {\delta}_{1}=\sqrt{\frac{5}{8}}.$

For

${\alpha}_{1}={\alpha}_{2}$, a computation shows that

${\delta}_{1}=\frac{1}{3}\sqrt{6}=\sqrt{\frac{2}{3}}.$

□

We conclude that the condition number is very close to its ideal unit value 1, leading to very few iterations. For instance, it suffices with at most 5 conjugate gradient iterations for a relative accuracy of 10^{−6}.

**Remark 3.1** High order implicit Runge-Kutta methods and their discretization error estimates can be derived using order tree methods as described in [1] and [10].

For an early presentation of implicit Runge-Kutta methods, see [2] and also [4], where the method was called global integration method to indicate its capability for large values of *m* to use few, or even just one, time discretization steps. It was also shown that the coefficient matrix, formed by the quadrature coefficients had a dominating lower triangular part, enabling the use of a matrix splitting and Richardson iteration method. It can be of interest to point out that the Radau method for $m=2$ can be described in an alternative way, using Radau quadrature for the whole time step interval and combined with a trapezoidal method for the shorter interval.

Namely, let

$\frac{du}{dt}+f(t,u)=0$,

${t}_{k-1}<t<{t}_{k}$. Then Radau quadrature on the interval

$({t}_{k-1},{t}_{k})$ has quadrature points

${t}_{k-1}+\tau /3$,

${t}_{k}$, and coefficients

${b}_{1}=3/4$,

${b}_{2}=1/4$, which results in the relation

${\tilde{u}}_{1}-{\tilde{u}}_{0}+\frac{3\tau}{4}f({\tilde{t}}_{1/3},{\tilde{u}}_{1/3})+\frac{\tau}{4}f({\tilde{t}}_{1},{\tilde{u}}_{1})=0,$

where ${\tilde{u}}_{1}$, ${\tilde{u}}_{1/3}$, ${\tilde{u}}_{0}$ denote the corresponding approximations of *u* at ${\tilde{t}}_{1}\doteq {t}_{k-1}+\tau $ and ${\tilde{t}}_{1/3}={t}_{k-1}+\tau /3$ and ${t}_{k-1}$, respectively.

This equation is coupled with an equation based on quadrature

$u({t}_{k-1}+\tau /3)-u({t}_{k-1})+{\int}_{{t}_{k-1}}^{{t}_{k}}f(t,u)\phantom{\rule{0.2em}{0ex}}dt-{\int}_{{t}_{k-1}+\tau /3}^{{t}_{k}}f(t,u)\phantom{\rule{0.2em}{0ex}}dt=0,$

which, using the stated quadrature rules, results in

${\tilde{u}}_{1/3}-{\tilde{u}}_{0}+\frac{3\tau}{4}f({\tilde{t}}_{1/3},{\tilde{u}}_{1/3})+\frac{\tau}{4}f({\tilde{t}}_{1},{\tilde{u}}_{1})-\frac{1}{2}\frac{2\tau}{3}[f({\tilde{t}}_{1/3},{\tilde{u}}_{1/3})+f({\tilde{t}}_{1},{\tilde{u}}_{1})]=0$

that is,

${\tilde{u}}_{1/3}-{\tilde{u}}_{0}+\frac{5\tau}{12}f({\tilde{t}}_{1/3},{\tilde{u}}_{1/3})-\frac{\tau}{12}f({\tilde{t}}_{1},{\tilde{u}}_{1})=0.$

**Remark 3.2** The arising system in a high order method involving $q\ge 2$ quadratic polynomial factors, can be solved sequentially in the order they appear. Alternatively (see, *e.g.* [11], Exercise 2.31), one can use a method based on solving a matrix polynomial equation, ${\mathcal{P}}_{2q}(A)\mathbf{x}=\mathbf{b}$ as $\mathbf{x}={\sum}_{k=1}^{q}\frac{1}{{\mathcal{P}}_{2q}^{\mathrm{\prime}}({r}_{k})}{\mathbf{x}}_{k}$, ${\mathbf{x}}_{k}={(A-{r}_{k}I)}^{-1}\mathbf{b}$, where ${\{{r}_{k}\}}^{2q}$, is the set of zeros of the polynomial and it is assumed that *A* has no eigenvalues in this set. (This holds in our applications.) Then, combining pairs of terms corresponding to complex conjugate roots ${r}_{k}$, quadratic polynomials arise for the computation of the corresponding solution components. It is seen that in this method, the solution components can be computed concurrently.

**Remark 3.3** Differential algebraic equations (DAE) arise in many important problems; see, for instance [

10,

12]. The simplest example of a DAE takes the form

$\{\begin{array}{c}\frac{du}{dt}=f(t,u,v),\hfill \\ g(t,u,v)=0,\phantom{\rule{1em}{0ex}}t>0,\hfill \end{array}$

with

$u(0)={u}_{0}$,

$v(0)={v}_{0}$ and it is normally assumed that the initial values satisfy the constraint equation,

*i.e.* $g(0,{u}_{0},{v}_{0})=0.$

If

$det(\frac{\partial g}{\partial v})\ne 0$ in a sufficiently large set around the solution, one can formally eliminate the second part of the solution to form a differential equation in standard form.

$\frac{du}{dt}=f(t,u,v(u)),\phantom{\rule{1em}{0ex}}t>0,u(0)={u}_{0}.$

Such a DAE is said to have index one, see

*e.g.* [

13]. It can be seen to be a limit case of the system

$\{\begin{array}{c}\frac{du}{dt}=f(t,u,v),\hfill \\ \frac{du}{dt}=\frac{1}{\epsilon}g(t,u,v),\hfill \end{array}$

where $\epsilon >0$ and $\epsilon \to 0$.

Hence, such an DAE can be considered as an infinitely stiff differential equation problem. For strongly or infinitely stiff problems, there can occur an order reduction phenomenae. This follows since some high order error terms in the error expansion (*cf.* Section 5) are multiplied with (infinitely) large factors, leading to an order reduction for some methods. Heuristically, this can be understood to occur for the Gauss integration form of IRK but does not occur for the stiffly stable variants, such as based on the Radau quadrature. For further discussions of this, see, *e.g.* [10, 13].