In order to simplify the notation and taking into account that most powerful results have been derived for scalar conservation laws in one spatial dimension, we will restrict ourselves to the initial value problem given by the equation

with the initial condition .

In hydrodynamic codes based on finite difference or finite volume techniques, equation (81) is solved on a discrete numerical grid with

and

where
and
are the time step and the zone size, respectively. A difference
scheme is a time-marching procedure allowing one to obtain
approximations to the solution at the new time,
, from the approximations in previous time steps. The quantity
is an approximation to
but, in the case of a conservation law, it is often preferable
to view it as an approximation to the average of
*u*
(*x*,
*t*) within a zone
(i.e., as a zone average), where
. Hence

which is consistent with the integral form of the conservation law.

Convergence under grid refinement implies that the global error , defined as

tends to zero as . For hyperbolic systems of conservation laws methods in conservation form are preferred as they guarantee that if the numerical solution converges, it converges to a weak solution of the original system of equations (Lax-Wendroff theorem [95]). Conservation form means that the algorithm can be written as

where
*q*
and
*r*
are positive integers, and
is a consistent (i.e.,
) numerical flux function.

The Lax-Wendroff theorem cited above does not establish whether the method converges. To guarantee convergence, some form of stability is required, as for linear problems (Lax equivalence theorem [154]). In this context the notion of total-variation stability has proven to be very successful, although powerful results have only been obtained for scalar conservation laws. The total variation of a solution at , TV(), is defined as

A numerical scheme is said to be TV-stable, if TV() is bounded for all at any time for each initial data. One can then prove the following convergence theorem for non-linear, scalar conservation laws [96]: For numerical schemes in conservation form with consistent numerical flux functions, TV-stability is a sufficient condition for convergence.

Modern research has focussed on the development of high-order, accurate methods in conservation form, which satisfy the condition of TV-stability. The conservation form is ensured by starting with the integral version of the partial differential equations in conservation form (finite volume methods). Integrating the PDE over a finite spacetime domain and comparing with (86), one recognizes that the numerical flux function is an approximation to the time-averaged flux across the interface, i.e.,

Note that the flux integral depends on the solution at the zone interface, , during the time step. Hence, a possible procedure is to calculate by solving Riemann problems at every zone interface to obtain

This is the approach followed by an important subset of shock-capturing methods, called Godunov-type methods [74, 48] after the seminal work of Godunov [66], who first used an exact Riemann solver in a numerical code. These methods are written in conservation form and use different procedures (Riemann solvers) to compute approximations to . The book of Toro [176] gives a comprehensive overview of numerical methods based on Riemann solvers. The numerical dissipation required to stabilize an algorithm across discontinuities can also be provided by adding local conservative dissipation terms to standard finite-difference methods. This is the approach followed in the symmetric TVD schemes developed in [38, 156, 197].

High-order of accuracy is usually achieved by using conservative monotonic polynomial functions to interpolate the approximate solution within zones. The idea is to produce more accurate left and right states for the Riemann problem by substituting the mean values (that give only first-order accuracy) by better representations of the true flow near the interfaces, let say , . The FCT algorithm [20] constitutes an alternative procedure where higher accuracy is obtained by adding an anti-diffusive flux term to the first-order numerical flux. The interpolation algorithms have to preserve the TV-stability of the scheme. This is usually achieved by using monotonic functions which lead to the decrease of the total variation (total-variation-diminishing schemes, TVD [72]). High-order TVD schemes were first constructed by van Leer [177], who obtained second-order accuracy by using monotonic piecewise linear slopes for cell reconstruction. The piecewise parabolic method (PPM) [33] provides even higher accuracy. The TVD property implies TV-stability, but can be too restrictive. In fact, TVD methods degenerate to first-order accuracy at extreme points [133]. Hence, other reconstruction alternatives have been developed where some growth of the total variation is allowed. This is the case for the total-variation-bounded (TVB) schemes [162], the essentially non-oscillatory (ENO) schemes [73] and the piecewise-hyperbolic method (PHM) [105].

Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
http://www.livingreviews.org/lrr-1999-3
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