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\begin{center}
  {\Large\sc Fonds National de la Recherche Scientifique}

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  {\large\bf Groupe de Contact en Analyse Num\'erique}

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 {\Large Numerical methods in ordinary and partial differential
         equations}

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  {\bf Vendredi 5 d\'ecembre 1997,\ \ \  Friday December 5, 1997}

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 {\large Facult\'e Polytechnique de Mons}

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   Laboratoire d'Automatique,\qquad   boulevard Dolez 31, Mons


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\bigskip\bigskip

 \noindent {\bf 9.30-10.00} Accueil  \qquad\qquad Welcome.

 \medskip
 \noindent {\bf 10.00-11.00} W. Schiesser (Lehigh University, Bethlehem,
  USA)

 {\it Space and time adaptivity
  in the numerical integration of partial differential equations.}

 \medskip
 \noindent {\bf 11.00-11.30} Caf\'e\ \ \ coffee
 \medskip

 \noindent {\bf 11.30-12.00}  Tanja Van Hecke (Universiteit Gent,
   Vakgroep Toegepaste Wiskunde en Informatica)

   {\it Mono-implicit Runge-Kutta-Nystr\"om Methods : efficient IRKN
   methods.}

 \noindent {\bf 12.00-12.30}  Jo Simoens (K.U.Leuven Department
  of Computer Science)

  {\it Fast Direct Solvers as Preconditioner
          in Waveform Relaxation.}

 \bigskip

 \noindent {\bf 12.30-14.30} Repas de midi (lunch).

 \bigskip

\noindent {\bf 14.30-15.00} Bart Sijnave  (Vakgroep Toegepaste Wiskunde
 en Informatica,   Universiteit Gent)

  {\it Continuation of codimension 2 equilibrium bifurcations
   in {\sc CONTENT}.}

 \medskip

 \noindent {\bf 15.00-15.15} break

 \medskip

 \noindent {\bf 15.15-16.15}  P. Zegeling (Math. Dept., Utrecht University,
         The Netherlands)

    {\it An adaptive grid method
  based on smoothed equidistribution and its application to PDEs with
  higher-order derivatives.}

 \medskip
 \noindent {\bf 16.15-16.30} Caf\'e\ \ \ coffee
 \medskip

 \noindent {\bf 16.30-17.00} G\'erard Degrez (von Karman Institute)

   {\it Implicit upwind residual distribution schemes for
    Euler/Navier-Stokes equations.}

 \noindent {\bf 17.00-17.30} Alain Vande Wouwer (Facult\'e Polytechnique
       de Mons)

  {\it Numerical Experiments with the Moving Finite Element Method}

\newpage

\begin{center}

     {\Large\sc Space and Time Adaptivity in the Numerical Integration
                       of Partial Differential Equations}

 \bigskip
                   {\large\sc  W. E. Schiesser   }

\bigskip
                              Lehigh University   \\
                          Bethlehem, PA  18015  USA

                               wes1@lehigh.edu
\end{center}

        An essential aspect of the numerical integration of partial
        differential equations (PDEs) is the monitoring and control
        of numerical errors.  In this paper, we review three basic
        approaches to adaptive error control (for discretization
        errors in space and time):

         \begin{enumerate}

           \item $h$ refinement: Variation of the space and time
                integration intervals.

           \item  $p$ refinement: Variation in the order of the space
                and time approximations.

           \item  $r$ refinement: Adaptive movement of the grid points
                to regions of large variations in the solution.
          \end{enumerate}

        These three strategies can be implemented manually (by observing
        the solution and making adjustments), or automatically (by having
        the computer code make the adjustments).

        We illustrate these strategies within the method of lines (MOL),
        which is a flexible approach to the numerical integration of
        systems of ordinary differential equations (ODEs),
        {differential-algebraic} equations (DAEs) and PDEs.

        Some representative numerical methods will be reviewed and illustrated
        through computer codes for: (a) a system of two initial-value ODEs
        of arbitrary stiffness and (b) a special case of the Einstein field
        equations of general relativity.

        The talk will conclude with a brief discussion of the software
        available for MOL analysis, including a set of Matlab ``m'' files
        and the Fortran code for the Einstein field equations that will be
        provided to the attendees.

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\begin{center}
{\Large\sc Mono-implicit Runge-Kutta-Nystr\"om Methods :
    efficient IRKN methods}

 \bigskip

{\large\sc  \underline{T. Van Hecke}}\,,  Research Assistent
          of the University of Gent, \\
  {\large\sc G. Vanden Berghe,  M. Van Daele} \\
   and {\large\sc H. De Meyer}\,, Research Director at the
National Fund for Scientific Research (N.F.W.O. Belgium)

\bigskip
{\em Vakgroep Toegepaste Wiskunde en Informatica},
{\em Universiteit--Gent} \\
{\em Krijgslaan 281 -- S9, B9000 Gent, Belgium }\\
{\em e-mail : Tanja.VanHecke@rug.ac.be}
\end{center}

Implicit Runge-Kutta-Nystr\"om methods (IRKN) are
determined by the Butcher tableau
\[
\begin{tabular}{c|c}
$c$ & A$\phantom{\int_0^1}$\\ \hline  & $\bar{b}^T$
$\phantom{\int_0^{\displaystyle{1}}}$
\\ &
 $b^T$ $\phantom{\int_0^1}$
\end{tabular}\]
with $c,\bar{b},b \in \rm{I\!R}^{s \times 1}$ and $A \in \rm{I\!R}^{s
\times s}$.
IRKN methods are expensive and explicit RKN methods have bad stability
properties. We consider the subclass of mono-implicit methods (MIRKN)
for which a strictly lower triangular matrix $X \in
\rm{I\!R}^{s\times s}$ and a vector $v \in \rm{I\!R}^{s \times 1}$
exist such
that the Butcher matrix $A$ can be written as
\[ A = X + v \bar{b}^T+w b^T\,.\]
Moreover, since we aim at the construction of P-stable
MIRKN methods,  we require that the $2 \times 2$
matrix stability function
\[ M(z) =
\left[ \begin{array}{ll}
1 + z \bar{b}^T (I-z A)^{-1} e & 1 + z
\bar{b}^T (I-z A)^{-1} c \\[.3cm]
z b^T (I-z A)^{-1} e & 1 + zb^T (I-zA)^{-1} c
\end{array}\right] \,, \label{stab2} \]
%whereby $e \in \Real^{s \times 1}$ denotes the vector with unit
whereby $e \in \rm{I\!R}^{s \times 1}$ denotes the vector with unit
entries,  satisfies for all real $z < 0$ both the conditions
det$M(z) = 1$ and (Tr$M(z))^2 < 4$.

An analysis is made of P-stable FSAL methods with $v=c$.
It will also be shown how the required LU-decomposition can be
simplified.
Symplectic methods are discussed within the class of MIRKN methods.



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\begin{center}

{\Large\sc Fast Direct Solvers as Preconditioner\\
in Waveform Relaxation}

\bigskip
{\large\sc Jo Simoens}

\bigskip
Katholieke Universiteit Leuven\\
Department of Computer Science\\
Celestijnenlaan 200A\\
B-3001 Heverlee\\
\texttt{\small(jo.simoens@cs.kuleuven.ac.be)}

\end{center}

Waveform relaxation is an iterative technique
for solving large systems of ODEs.
It is the continuous-in-time analogue of
iterative methods for stationary problems based on matrix splitting,
such as the Jacobi and Gauss--Seidel iterations.
In this talk we consider its application for solving
semi-discretised parabolic initial-boundary value problems.

The waveform relaxation method parallellises very well
if the preconditioner is highly parallellisable.
We discuss the use as preconditioner of direct parabolic PDE solvers
with known optimal parallel complexity.
These preconditioners are derived from direct solvers for elliptic PDEs,
and use the FFT in the spatial dimensions and cyclic reduction
in the time dimension.
Fine and coarse-grain variants of the parallel algorithm are presented.

The convergence rate of the resulting waveform relaxation iteration
is shown to be independent of the mesh size
used in the spatial discretisation.
We conclude by comparing the method with some other
waveform relaxation and multigrid methods for parabolic problems.



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\begin{center}


{\Large\sc Continuation of codimension 2 equilibrium bifurcations
 in {\sc CONTENT}}

\bigskip

{\large\sc\underline{B.~Sijnave}\,, W.~Govaerts} (Fund for Scientific Research
(F.W.O. Belgium)

  and {\large\sc Yu.~A.~Kuznetsov} \ (CWI, Amsterdam (The Netherlands))

\bigskip
{\normalsize \em Vakgroep Toegepaste Wiskunde en Informatica,
Universiteit--Gent} \\
{\normalsize \em Krijgslaan 281 -- S9, B9000 Gent, Belgium} \\
{\normalsize \em e-mail : Bart.Sijnave@rug.ac.be}
\rm

\end{center}

In this talk, we discuss recent extensions to {\sc CONTENT}, an
interactive software package developed at CWI (Amsterdam) by
Yu.~A.~Kuznetsov and V.~V.~Levitin, to compute and pathfollow
numerically all codimension 2 bifurcations of the solutions of the
equilibrium equations $F(u,\alpha) = 0$ associated with the dynamical
system
\[
 \dot{u} = F(u,\alpha) \qquad u,F(u) \in \R^n, \alpha \in \R^m\ .
\]
We will deal with Bogdanov-Takens points ({\tt BT}), Zero-Hopf points
({\tt ZH}), Double Hopf points ({\tt DH}) and Generalized Hopf points
({\tt GH}) which can be found while following a curve of single Hopf
points and also with cusp points ({\tt CP}) that can be found on limit
point curves. Also, detection of several codimension 3 bifurcations will
be discussed.

We illustrate the use of this software with several model computations
for all types of the above mentioned bifurcations. One of the presented
models is a realistic model of a neuron, developed at Cornell University
by J.~Guckenheimer and coworkers.

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\begin{center}

{\Large\sc An adaptive grid method based on smoothed equidistribution and its
application to PDEs with higher order derivatives.}
\bigskip

{\large\sc Dr. P. A. Zegeling}

\bigskip

Dept. of Mathematics, University of Utrecht \\
P.O. Box 80.010, 3508 TA Utrecht, The Netherlands \\
e-mail: zegeling@math.ruu.nl  \\
WWW: http://www.math.ruu.nl/people/zegeling/

\end{center}

Abstract: Traditional numerical techniques to solve time-dependent PDEs
integrate on a uniform spatial grid that is kept fixed on the entire time
interval. If the solutions have regions of high spatial activity, a standard
fixed-grid technique is computationally inefficient, since to afford an
accurate numerical approximation, it should contain, in general, a very
large number of grid points.

The grid on which the PDE is discretized then needs to be locally refined.
Moreover, if the regions of high spatial activity are moving in time, like
for steep moving fronts in reaction-diffusion or hyperbolic equations, then
techniques are needed that also adapt (move) the grid in time: continuously
deforming grid methods, which are also denoted by the term r-refinement.

In this talk I will describe an adaptive moving-grid method that is based on
an equidistribution
principle supplied with smoothing both in the time and space direction. In
one space dimension this is rather straightforward and it can theoretically
be shown that the ratios of adjacent spatial grid cells are forced to be
bounded from below and above, due to the smoothing. In two dimensions, an
adaptive moving grid can be defined by applying the 1D-principle along the
two coordinate directions.
Numerical results will be shown in one and two space dimensions for, among
others, reaction-diffusion models and higher-order PDEs in time and space, such
as wave equations, KWdV-equation and the Extended Fisher-Kolmogorov PDE model.

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\begin{center}
   {\Large\sc Implicit upwind residual distribution schemes for
    Euler/Navier-Stokes equations.}

\bigskip

{\large\sc G\'erard Degrez}

\bigskip
     Von Karman Institute for Fluid Dynamics,\\
    72, Chaussee de Waterloo Waterloose steenweg 72\\
     1640 Rhode-Saint-Genese 1640 Sint-Genesius-Rode Belgium

\end{center}

A review is made of implicit residual distribution schemes for the solution
of compressible flows on unstructured grids which have been extensively
studied at the von Karman Institute over the past few years.

Space discretization is considered first. The main motivation for the
development of upwind residual distribution discretizations is to genuinely
account for the multidimensional flow physics, contrary to state-of-the-art
finite volume solvers which are based on 1D Riemann solvers. The basic
principles of residual distribution schemes are explained in detail for the
linear scalar advection equation. It is shown in particular that more
classical schemes such as the SUPG finite element and Lax-Wendroff schemes
can be viewed as residual distribution schemes. Then, their generalization
to non-commuting hyperbolic systems is presented. The application of upwind
residual schemes to the non-linear Euler equations of gas dynamics requires
two additional ingredients, i.e. suitable decompositions of the system and
a conservative linearization, which are discussed next. Finally, extension
to viscous flow problems is achieved by a Galerkin finite element
discretization of the viscous terms.

Having selected the space discretization scheme, there remains to specify
the iterative solution strategy for the resulting set of non-linear
algebraic equations. We consider here an implicit time-stepping/damped
Newton strategy. It is shown that, thanks to the compactness of the space
discretization, it is possible to compute the Jacobian matrix needed by the
damped Newton algorithm economically both in terms of CPU time and storage.
Linear systems are solved iteratively using preconditioned Krylov subspace
algorithms (GMRES, BiCG-STAB).
Finally, the presentation concludes with illustrative applications to a
wide range of inviscid, laminar and turbulent flows.



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\begin{center}

{\Large\sc
Numerical Experiments with the Moving Finite Element Method }

\bigskip

{\large\sc Alain Vande Wouwer}

\bigskip
  Facult\'e Polytechnique de Mons

  Automatic Control Lab

\end{center}

\bigskip

In this talk, the moving finite element and the gradient weighted moving
finite element methods for solving partial differential equations in one
space dimension are briefly described. Several test-examples from science
and engineering are used to illustrate the method features, including the
selection of an initial node distribution, the parameter tuning and the use
of matrix preconditioning.


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