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 conditionIn hydrodynamic codes based on finite difference
or finite volume techniques, Equation (92) is solved on a
discrete numerical grid
with
Convergence under grid refinement implies that
the global error , defined as
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 [243]). 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
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 Equation (97
), one recognizes that
the numerical flux function
is an
approximation to the time-averaged flux across the interface,
i.e.,
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’s say
,
. The FCT
algorithm [33] 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 [121]). High-order TVD schemes
were first constructed by van Leer [282], who obtained
second-order accuracy by using monotonic piecewise linear slopes
for cell reconstruction. The piecewise parabolic method
(PPM) [60] 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 [215]. 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 [258], the essentially
non-oscillatory (ENO) schemes [120] and the
piecewise-hyperbolic method (PHM) [175].
There are several strategies to extend HRSC
methods to more than one spatial dimension. A brief summary of
these strategies can be found in LeVeque’s book [158] (see also [161]). The simplest strategy
is dimensional splitting, where the differential operators along
the spatial directions are applied in successive steps (fractional
step methods). Second order in time is achieved when one permutes
cyclically the order in which the directional (i.e., 1D) operators
are applied (Strang splitting [270]). In semi-discrete
methods (method of lines), the process of discretization proceeds
in two stages. First only operators involving spatial derivatives
are discretized, leaving the problem continuous in time. This gives
rise to a system of ordinary differential equations (in time) which
can be integrated by any ODE solver. In the method of lines
approach, the numerical fluxes across cell interfaces are computed
in all two or three spatial directions, before they are
simultaneously applied to advance the equations. Particularly of
interest are TVD Runge-Kutta time discretization
algorithms [259, 260], which preserve the TVD
properties of the algorithm at every substep. A third approach
relies on unsplit methods, where the different spatial directions
are also advanced simultaneously as in the semi-discrete methods.
However, the extension of unsplit methods to second-order accuracy
requires incorporating not only slopes in the normal direction (as
in one-dimensional or split algorithms), but also cross-derivatives
arising from the multi-dimensional Taylor series expansion. Good
examples of genuinely multi-dimensional upwind methods for
hyperbolic conservation laws (using slightly different strategies)
are those described in [58, 159
]. In [58] the algorithm proceeds
in two steps. First, interface values are interpolated, using
information from all orthogonal directions. Secondly, the Riemann
problems defined by these interface values are solved. The
algorithm proposed in [159] first solves the Riemann
problem, and then distributes the information to the appropriate
directions.