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].
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Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller http://www.livingreviews.org/lrr-1999-3 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |