252
5. Finite-Volume Methods
5.2.2 1VD Methods
First-order methods do not provide a particularly efficient way to obtain accurate numerical solutions; better results can often be obtained using higher-order
schemes. Although they are not monotone, many of these schemes satisfy the
weaker stability condition that they are total variation nonincreasing, The total
variation of a one-dimensional grid-point function is defined as
N-I
TV(rp) = L Irpj+1 - rpjl,
j=1
where N is the total number of grid points in the numerical domain . The total
variation of a continuous function on the interval [a, b] may be defined in an
analogous manner as the supremum over all possible subdivisions of the domain
a = XI < X2 < ... < XN = b of
N-J
L 1t/J(xj+l) - t/J(Xj) I,
j=1
E--+O
E
or equivalently as
11
TV(t/J) = limsup -
-00
00
It/J(x + E) - t/J(X)Idx = O.
A numerical method is total variation nonincreasing if
(5.20)
Although slightly imprecise, it is common practice and easier on the tongue to refer to a method that is total variation nonincreasing as total variation diminishing,
or TVD. This convention will be followed in the remainder of this book, so that
(5.20) is the working definition of a TVD method.
Solutions to a consistent finite-difference method in conservation form are
guaranteed to converge to weak solutions of the exact conservation law whenever the scheme is TVD. The nature of this convergence is, however, complicated
by the fact that there may be several nonunique weak solutions to a given conservation law. If the scheme is TVD, the infimum, over the set of all possible
weak solutions, of the difference between the numerical solution and each weak
solution is guaranteed to go to zero as /),.X -+ 0 and /),.t -+ 0, but a sequence of numerical solutions computed with successively smaller values of /),.X and /),.t need
not smoothly converge to any particular weak solution (LeVeque 1992, p. 164).
Of course, the goal is to obtain an approximation that converges to the entropyconsistent solution, and this is generally accomplished by demanding that every
approximate solution also satisfy a discrete form of the entropy condition .
The family of monotone finite-volume schemes is a subset of the family ofTVD
schemes, which are in turn a subset of an even more general dass of monotonicitypreserving methods (Harten 1983). A method is monotonicity-preserving if it will
5. Finite-Volume Methods
5.2.2 1VD Methods
First-order methods do not provide a particularly efficient way to obtain accurate numerical solutions; better results can often be obtained using higher-order
schemes. Although they are not monotone, many of these schemes satisfy the
weaker stability condition that they are total variation nonincreasing, The total
variation of a one-dimensional grid-point function is defined as
N-I
TV(rp) = L Irpj+1 - rpjl,
j=1
where N is the total number of grid points in the numerical domain . The total
variation of a continuous function on the interval [a, b] may be defined in an
analogous manner as the supremum over all possible subdivisions of the domain
a = XI < X2 < ... < XN = b of
N-J
L 1t/J(xj+l) - t/J(Xj) I,
j=1
E--+O
E
or equivalently as
11
TV(t/J) = limsup -
-00
00
It/J(x + E) - t/J(X)Idx = O.
A numerical method is total variation nonincreasing if
(5.20)
Although slightly imprecise, it is common practice and easier on the tongue to refer to a method that is total variation nonincreasing as total variation diminishing,
or TVD. This convention will be followed in the remainder of this book, so that
(5.20) is the working definition of a TVD method.
Solutions to a consistent finite-difference method in conservation form are
guaranteed to converge to weak solutions of the exact conservation law whenever the scheme is TVD. The nature of this convergence is, however, complicated
by the fact that there may be several nonunique weak solutions to a given conservation law. If the scheme is TVD, the infimum, over the set of all possible
weak solutions, of the difference between the numerical solution and each weak
solution is guaranteed to go to zero as /),.X -+ 0 and /),.t -+ 0, but a sequence of numerical solutions computed with successively smaller values of /),.X and /),.t need
not smoothly converge to any particular weak solution (LeVeque 1992, p. 164).
Of course, the goal is to obtain an approximation that converges to the entropyconsistent solution, and this is generally accomplished by demanding that every
approximate solution also satisfy a discrete form of the entropy condition .
The family of monotone finite-volume schemes is a subset of the family ofTVD
schemes, which are in turn a subset of an even more general dass of monotonicitypreserving methods (Harten 1983). A method is monotonicity-preserving if it will