Computational Relativity The largest field of enquiry historically has been the field of exact solutions of Einstein's field equations. The Einstein field equations constitute an extremely complicated set of non-linear partial differential equations. It came as something of a surprise when Schwarzschild found an exact solution within a year of the field equations being published. In the ensuing years very few exact solutions were found until new invariant techniques were introduced in the mid 1960s. This led to an explosion of exact solutions being discovered. Many of the calculations associated with exact solutions are straightforward but extremely long and complicated to perform which can easily lead to errors. Around the mid 1960s the field of Algebraic Computing in General Relativity came into existence and it soon became possible to undertake calculations with computers that would take more than a lifetime to complete by hand. An example (and one which has been used as a standard for comparing algebraic computing systems) is the calculation of the Ricci tensor for Bondi's radiating metric and was first undertaken by Ray d’Inverno using his algebraic computing system LAM (which eventually developed into the system SHEEP). The original hand calculation was undertaken over a period of some six months and now takes less than a second on a reasonable spec PC. Part of the SHEEP output for the famous Bondi radiating metric. The user asks SHEEP to make and write (wmake) the line element which is called ds2 and then to make the Ricci tensor and the first 23 terms in the output are displayed above. The advent of Algebraic Computing led to a new attack on the famous equivalence problem of General Relativity, namely: given two metrics, does there exist a local transformation which transforms one into the other? Cartan showed in some classic work that the problem can be solved but depends on computing the 10th covariant derivative of the Riemann tensor of each metric. Even with modern computer algebra systems this is out of the question. The work of Karlhede significantly improved the situation. Karlhede's approach provides an invariant classification of a metric. Thus, if two metrics have different classifications then they are necessarily inequivalent, whereas if they have the same classification then they are candidates for equivalence. The problem then reduces to solving four algebraic equations. Karlhede's algorithm reduces the derivative bounds significantly below the original 10 of Cartan. There is particular interest in the Southampton group in reducing the bounds which might occur in various cases, for if they can be reduced sufficiently then it becomes possible to use algebraic computing systems to classify exact solutions. The advent of the Karlhede classification algorithm has lead to the setting up of a computer database of exact solutions through collaborative work between UERJ, Rio de Janeiro, Brazil and Southampton. At present some 200 metrics exist in the computer database of exact solutions. The ultimate hope is that it will contain all known solutions, fully documented and classified. Then any `newly' discovered solution can be compared with the contents of the database and, if indeed it is new, then the database can be updated ccordingly. Were this to be fully realised then it would provide a valuable resource for the international community of relativists

While several hundred exact solutions to the field equations are known to exist, few of these solutions describe physically relevant situations. If we want to solve the equations in scenarios of astrophysical interest, such as the birth of a neutron star in a supernova or the collision of two black holes in a binary system, we need to perform large scale numerical simulations. In the last ten years or so, there has been an international effort to develop Numerical Relativity to a level where simulations can provide reliable information about the most violent events in the Universe. This development was largely motivated by the construction of new generations of gravitational wave detectors. In addition, it is only recently that computers have become powerful enough that one can hope to achieve the desired computational precision (in a reasonable computing time). After all, Einstein's equations are extremely complicated (they involve over 100,000 terms in the general case) and solving them numerically makes enormous demands on the processing power and memory of a computer. Because of this, the development of Numerical Relativity has proceeded in stages first solving 1-dimensional problems (that is 1 spatial dimension), then 2-dimensional problems and only very recently have fully 3-dimensional problems been tackled. One of the major problems in NR is that there is no known local expression for gravitational waves - they can only be properly described asymptotically (that is out at infinity). In most previous work, numerical simulations are carried out on a central finite grid extending into the vacuum region surrounding the sources present and ad-hoc conditions are imposed at the edge of the grid to prevent incoming waves (which would be unphysical). Unfortunately, these ad-hoc conditions themselves generate spurious reflected numerical waves. The basic formalism that the numerical codes employ is the Arnowitt-Deser-Misner 3+1 formalism which decomposes 4-dimensional space-time into a family of constant time 3-dimensional spatial slices. Although this formalism is well adapted to central regions it does not work well in asymptotic regions. Instead, the d’Inverno-Stachel-Smallwood 2+2 formalism decomposes space-time into two families of 2 dimensional spacelike surfaces. The formalism encompasses 6 different cases, one of which is called the null-timelike case in which one of the families can be taken to form null 3-surfaces and the other to form timelike 3-surfaces. Null surfaces are especially important in General Relativity because they are ruled by null geodesics and these are the curves along which gravitational information is propagated. The idea behind the Southampton CCM (Cauchy-Characteristic Matching) project is to combine a central 3+1 numerical code with an exterior null-timelike 2+2 code connected across a timelike interface residing in the vacuum. In addition, the exterior region is compactified so as to incorporate null infinity where gravitational radiation can be unambiguously defined. The Southampton project has to date used CCM to investigate cylindrically symmetric and axially symmetric systems as a prelude to investigating fully 3 dimensional systems. The significance of this work is that it leads to wave forms asymptotically, and it is these exact templates which are needed in the search for gravitational waves. A Teukolsky gravitational wave on a finite 3+1 grid instead of dissipating off of the grid leads to a small spurious reflection (see second diagram). The Southampton CCM code provides a global code in which the wave reaches infinity unaltered and where it can be unambiguously characterised Left: results from the US Binary Black Hole Grand Challenge supercomputer simulations of the coalescence of two black holes : the merger of the two horizons lead to the famous `pair of pants' picture. Right: NASA supported Netron Star Grand Challenge: the simulation of the merger of two Neutron stars emitting gravitational waves.