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50 Finite-Volume Methods
be obtained by averaging the velocities between time levels n and n + I, or by
extrapolating forward from time levels n and n-I (see Problem 11). A formula for
the approximation of advective fluxes in spatially varying nondivergent velocity
fields will be presented in Section 5.7.3.
5.6 Approximation with Local Polynomials
Formulae very similar to those obtained with the flux-limiter approach can be
derived by approximating the solution as the sum of piecewise-Iinear functions
defined over each grid cell and then computing the evolution of this piecewiselinear approximation over a time interval !i.t, e.g., van Leer (1974). Similar methods can be derived using other piecewise-continuous polynomials. The simplest
scheme, due to Godunov (1959), is obtained using piecewise-constant functions.
Greater accuracy was achieved by Colella and Woodward (1984) using piecewiseparabolic functions, The following sections will discuss piecewise-constant approximations to nonlinear one-dimensional scalar conservation laws of the form
(506) and piecewise-linear approximations to the constant-wind-speed advection
equation.
5.6.1 Godunov's Method
In Godunov's method, the gridpoint values at each individual time step are used
to define a piecewise-constant function such that
where t" = nSt and x
= x j +!:!..x/2. Using the function (P(x,' rn) as the initial
condition, an approximate solution to the original conservation law at r n +
1 may
be obtained by solving the Riemann problems associated with the discontinuities
in (P at the interface of each grid cell. The exact solution to these Riemann problems can be easily obtained for a scalar conservation law or for linear systems of
conservation laws, at least until the signals emanating from each interface begin
to interact,? The new solution at time rt>j+1 is defined to be the average of these
individual Riemann solutions over the jth grid cell,
7Sce LeVeque (1992) for a discussion of approximate teehniques for the solution of Riemann
problems involving nonlinear systems of conservation laws.
50 Finite-Volume Methods
be obtained by averaging the velocities between time levels n and n + I, or by
extrapolating forward from time levels n and n-I (see Problem 11). A formula for
the approximation of advective fluxes in spatially varying nondivergent velocity
fields will be presented in Section 5.7.3.
5.6 Approximation with Local Polynomials
Formulae very similar to those obtained with the flux-limiter approach can be
derived by approximating the solution as the sum of piecewise-Iinear functions
defined over each grid cell and then computing the evolution of this piecewiselinear approximation over a time interval !i.t, e.g., van Leer (1974). Similar methods can be derived using other piecewise-continuous polynomials. The simplest
scheme, due to Godunov (1959), is obtained using piecewise-constant functions.
Greater accuracy was achieved by Colella and Woodward (1984) using piecewiseparabolic functions, The following sections will discuss piecewise-constant approximations to nonlinear one-dimensional scalar conservation laws of the form
(506) and piecewise-linear approximations to the constant-wind-speed advection
equation.
5.6.1 Godunov's Method
In Godunov's method, the gridpoint values at each individual time step are used
to define a piecewise-constant function such that
where t" = nSt and x
= x j +!:!..x/2. Using the function (P(x,' rn) as the initial
condition, an approximate solution to the original conservation law at r n +
1 may
be obtained by solving the Riemann problems associated with the discontinuities
in (P at the interface of each grid cell. The exact solution to these Riemann problems can be easily obtained for a scalar conservation law or for linear systems of
conservation laws, at least until the signals emanating from each interface begin
to interact,? The new solution at time rt>j+1 is defined to be the average of these
individual Riemann solutions over the jth grid cell,
7Sce LeVeque (1992) for a discussion of approximate teehniques for the solution of Riemann
problems involving nonlinear systems of conservation laws.