NASA/CR-2002-211959 ICASE
Report
No. 2002-42
Local Discontinuous Differential
Equations
Jue Yan and Chi-Wang Brown
University,
November
2002
Galerkin
with Higher
Shu
Providence,
Methods
Rhode Island
for Partial
Order Derivatives
The
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NASA/CR-2002-211959 ICASE
Report
No.
2002-42
Local Discontinuous Differential Jue
Yan and
Brown
Equations
Chi-Wang
University,
Galerkin
Methods
with Higher
for Partial
Order Derivatives
Shu
Providence,
Rhode
Island
ICASE NASA
Langley
Research
Hampton,
Virginia
Operated
by Universities
Center
Space
Research
Association
Prepared for Langley Research under Contract NAS 1-97046
November
2002
Center
Available
from the following:
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for AeroSpace Drive
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(301) 621-0390
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(CASI)
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Technical
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Service
(NTIS)
LOCAL
DISCONTINUOUS
GALERKIN
EQUATIONS
WITH
HIGHER
JUE
Abstract.
In this
for solving space
time
local
second
correct
derivatives
and
technique,
with
are shown
originally
negligible
words,
stability,
error
Subject
discontinuous estimate,
derivative cost,
for linear
type
nonlinear
we present
We then
equations
problems.
estimates,
Numerical
in effectively
in-
develop
new
involving
fourth
we present
Preliminary
numer-
on a post-processing on the local discontin-
experiments
doubling
as well as some
equations
new methods
new results
error
and multiple
diffusion
derivatives.
For these
derivatives.
cases,
in one
methods
show that
the rate
nonlinear
this
of convergence
problems,
with
a local
Introduction.
Applied
we review
partial
We consider
A hyperbolic
method,
partial
and Numerical
In this paper
time dependent
dimensions.
derivatives.
Galerkin
differential
equations
with
higher
derivatives,
post-processing
classification.
ods for solving space
computational
third
negative-order
with higher
derivatives
for convection
bi-harmonic
Finally,
with good
order
derivatives.
for general
methods.
to equations
higher
Galerkin
mesh.
Key
1.
L 2 stability
new local discontinuous
involving
fifth
DIFFERENTIAL
_
methods
dependent
and prove
as well for the new higher additional
time
involving
for methods
applied
equations
for the
these
with
Galerkin
equations
to illustrate
designed
methods
works
uniform
methods
fluxes
equations
type
SHU
develop
PARTIAL
DERIVATIVES*
CHI-WANG
and
discontinuous
FOR
ORDER
AND
the existing
for KdV
differential
numerical
uous Galerkin technique
and
YAN f
differential
local
Galerkin partial
interface
ical examples
partial
We review
discontinuous
derivatives,
we review
dependent
dimensions.
volving
paper
METHODS
existing
differential
a sequence
conservation
Mathematics and develop
equations
new local discontinuous
of such partial
with higher differential
order
derivatives
equations
Galerkin
meth-
in one and multiple
with increasingly
higher
order
law d
Ut + E
fi(U)x,
= 0
(1.1)
i_1
is a partial
differential
equation
with
first derivatives.
The convection
d
d
Ut + E
f,(U)x,
where
(aij(U))
derivatives.
is a symmetric,
The general
KdV
type
E
i=1 *Research
supported and
Contract
by
ARO
grant
NAS1-97046
Hampton, VA 23681-2199, and AFOSR ¢Division of Applied Mathematics, SDivision
of Applied
Mathematics,
definite
(aij(U)U_)_:,
= 0
(1.2)
j=l
matrix,
is a partial
differential
equation
with
second
equation
Ut+Efi(U)x,+
NCC1-01035
E
i=l
semi-positive
equation
d
- E
i----1
diffusion
r_(U)
i=l DAAD19-00-1-0405, while
the
second
Egij(ri(U)x,)xj
=0
j=l NSF author
grants was
(1.3)
xi DMS-9804985 in residence
and
ECS-9906606,
ICASE,
NASA
at
NASA Langley
grant Brown
F49620-99-1-0077. University, Providence,
R!
02912.
E-mail:
yjue_cfm.brown.edu
Brown
University,
RI
02912.
E-mail:
shu_cfm.brown.edu
Providence,
Langley
Research
grant Center,
is a partial
differential
equation
with third
derivatives.
d
+ E(ai(Ux,)Ux,x,)x,x,
i=1
we just
differential
present
equation
(1.4)
with
as an example.
fourth
derivatives,
The following
differential
be presented. physical
equation
present
(1.5)
All these
The type of discontinuous finite
element
Runge-Kutta
time
derivatives section
review
paper
containing
applied.
in a series
equation
A naive
containing
higher
derivatives
method
on the
general
but
of papers
could
or higher
state
letters
be more
derivatives
counterparts,
U etc.
general
could
appear
to denote
also
often
in
the solutions
to
solutions.
developed
in this paper, with
using a discontinuous
explicit,
for the
nonlinearly
conservation
[8, 9, 6, 4, 10].
method
laws
(1.1)
We will briefly
[5], other
papers
in that
Galerkin
stable
review
as well as its implementation
paper
derivatives
to approximate
variables
is why the
method
first
local
and
high
order
containing
first
this
method
in
applications,
Springer
Shu [20] developed
In both
nonlinear
examples
were shown
reviewed
in sections
a "local"
volume,
method second
Rebay
we
and the
[11] and [20], suitable of the methods
to illustrate
the
stability
equation
we develop
(1.4) involving
fluxes
the new development
new local discontinuous derivatives,
and
Galerkin
and in section
of interface
of all the auxiliary
Shu
[11], for the by the
These
Yan and
(1.3) containing
were given,
for the linear
con-
successful
Later,
equation
internees
methods.
in later
Galerkin
of all the auxiliary
equations.
KdV type
estimates
with
in [11].
Navier-Stokes
of these
design
work was motivated
at element
to the
equations
the discontinuous
by Cockburn
for the general
as well as error
differential
The local solvability
Their
directly
in the computation
is the correct
method
at
[18, 12].
and local solvability
Galerkin
was developed
and accuracy
fourth
methods
stability
derivatives.
numerical
L 2 stability
paper
of such
be
"non-conforming"
nicely
apply
cannot
discontinuous
method
solution
partial
then
of the solution.
method
behaves
dependent
[1] for the compressible
Galerkin
Galerkin
to the exact
system,
methods
polynomials
This is a typical
which
errors
discontinuous
Galerkin
2.2 and 2.3, to motivate
3 of this
success
containing
of Bassi and
a method
to guarantee
the derivatives
a local discontinuous
derivatives.
type
(1.2)
for the
derivatives.
for time
Galerkin
of piecewise
of the discontinuous
into a first order
be designed
is called
discontinuous
equation
experiments
In section
fluxes must
discontinuous
consists
higher
yield
methods
the equation
introduced
diffusion
which
and has O(1)
Galerkin
A key ingredient
These
could
equation
discontinuous
system.
derivatives,
space,
application
second
is to rewrite
spatial
and careless
variables
bi-harmonic
steady
coupled
to handle
fluxes.
to provable
sixth
we use capital
enough
numerical
third
be more
(1.5)
the nonlinearity
with
we will discuss
of the
order
with the original
The idea of local
numerical
paper
variables
the solution
is not regular
elements.
but is inconsistent
The
again
independent
[3], the survey
higher
This is because
interfaces,
case in finite
vection
could
[12].
the element
heat
spatial
notes
where equations
the numerical
methods
description
to the lecture
For equations directly
In this
[19], were first
et al.
For a detailed
refer the readers
time
their
to denote
discretization
by Cockburn
2.1.
and
in the
the nonlinearity
=o
derivatives, Similar
Galerkin
approximation
(1.4)
i=1
fifth
applications.
and lower case letters
where
+
example.
equations,
and engineering
the PDEs
with
as an
-- 0
d
i=1
we just
equation
equation
c5 +
but
bi-harmonic
i=1
d
is a partial
dependent
d
Ut + Zf_(U)x,
is a partial
The time
cases.
results
which
led
Numerical
will be briefly
sections. methods
for the
time dependent
4 we do the same
thing
for the
partialdifferential equation(1.5)involvingfifth derivatives. Similarmethodscanbedesigned forwellposed partialdifferentialequations involvingevenhigherderivatives.Asbefore,wegiverecipesfor correct interelement
numerical
is independent suitable
fluxes
of the
for the
which
coefficients
so-called
terms
and hence
extremely
local and hence numerical
usual
and
proven)
piecewise
of this post-processing
technique
is sufficiently
and
smooth,
will achieve
the
convergence
of the
post-processing linear
order
L2-norm
post-processing.
with
higher
piecewise
(1.3),
polynomials
computational Concluding
linear
in Galerkin
the same
f'rom Ax k+l before strongly
suggests
of degree
k are used. accuracy
this
step
is applied
accuracy
enhancement
that
the
The
post-processing
in the
technique
thus
in [20] for the
only at the end of the computation
post-processing the order we apply
in this
paper,
for all these (in many
cases, cases
is fully
this
(1.5) where
Ax 2k+_)
for these
equations
norm,
taking
when
advantage
nonlinear
of
problems,
KdV equations.
only locally,
of
for the
PDE
negative-order
nonlinear
is
paper,
for some
and is applied
the
the solution
than
designed
can even be observed method
than
(1.4), and the linear
we have
of 2k + 1 or higher
Galerkin
5 of this
capability
methods
the
bigger
to Ax 2k+1 or higher
LDG
is locally
post-processing
is linear,
in [20] and
equations
mesh
The key ingredient
then
is usually
method
bi-harmonic
post-processing
The
estimate,
In section
Galerkin
are
laws (1.1) (theoretically
experiments.
(which
methods
We will provide
if the
thus if the problem
norm
enhancement
it to the local discontinuous
observed).
methods).
of convergence
that
(numerically
error
especially
to the higher
of Ax 2k+1 rather
conservation
negative-order
time dependent
have an order
To demonstrate
post-processing
error
We observe
This
derivatives
we also apply
has a good
convolution
in numerical
result
methods.
to an accuracy
estimate,
of the negative-order
of the
has been increased
after
method
error
are
Also, these
for linear
cases
to our new local discontinuous
fifth derivatives.
accuracy
this.
the
technique
(1.2)
is a negative-order
methods
coefficients
terms.
of these
is a local
solution
nonlinear
the
small
The stability
and easy for h-p adaptivity.
k are used,
equations
for some
of convergence
KdV like equations
containing
diffusion
accuracies
with
convection
in [7], which
of degree
methods.
hence
those
and accuracy
the
of the
derivatives, i.e.
derivative
the stability
was introduced
convection
in enhancing
order
implementations
to be able to recover polynomials
L 2 stability
problems,
by the first
to verify
proven
and for linear
even useful
of the higher
for parallel
technique
has been
Ax k+l when
efficient
nonlinear
dominated"
are dominated
examples
A post-processing uniform
in front
"convection
derivative
preliminary
lead to provable
As the
the additional
cost is negligible. remarks
summarizing
results
in this paper
and indicating
future
work are included
in section
6. 2.
Review
derivatives. conservation diffusion
laws
which
2.1.
discontinuous
section
(1.1) with
equations
For simplicity indicate
of the In this
(1.2)
first
with
of presentation, results
Laws.
the
second
methods
essential
derivatives,
the
derivatives
local and
for the general
let's
1,...,
N, with
xj+½
- xj_½.
introduce the
center
some
of the cell denoted
We will denote
Ax
= maxj
The
the KdV
Axj.
type
cases
version
=
second
(1.3) the
for the
with
methods.
and
methods
methods
equations
to present
first,
Galerkin
third
convection derivatives.
However,
laws
we will
(1.1) has the form
= O.
1 (xj_½
third for the
cases.
of the conservation
(2.1)
computational
by xj
with
Galerkin
multi-dimensional
The one dimensional
notations.
PDEs
of the discontinuous
discontinuous
Ut + I(U)x First
for
points
we will use one dimensional
are also valid
Conservation
Galerkin
we review
mesh
is given
+ xj+½)
If we multiply
(2.1)
and
by Ij the
=[xj_½,xj+½],
size of each
by an arbitrary
test
for j =
cell by Axj function
V(x),
=
integrate
over the interval
f This
the
UtVdx
is the starting
discontinuous test
Ij,
function
by parts,
+
flr
point
Galerkin
and integrate
for designing
method
we get
f(U(xj+½,
t))V(xj+½)
the discontinuous
Galerkin
[9, 6, 4, 10] can be described
V by piecewise
polynomials
f(U(xj_½,
of degree
t))V(xj_½)
method.
as follows:
at most
O.
The semi-discrete
we replace
k, and
=
denote
both
them
(2.2)
version
of the
the solution
by u and
U and
v.
That
is,
u, v E 1;A_ where IZA_ = {V : V is a polynomial With
this
xj+½,
as both
crucial
choice,
there
ingredient
choice
of discontinuity
f(u)
u and the
for the success
of the numerical at the
(information
is an ambiguity
the solution
in (2.2)
fluxes
to overcome
element
interfaces.
last
k for x E Ij, two
Galerkin
The
method
idea
j = 1,...,N}.
involving exactly
is to treat
by a single
these
the
boundary
values
at
boundary
points.
A
terms
monotone
laws is the
way, to utilize)
this
finite
volume
numerical
correct
ambiguity
by an upwinding
high resolution
valued
(2.3)
at these
for the conservation
say it in a positive
from the successful
j is given
terms
v are discontinuous
(or we could
borrowed
xj+ ½ for each
at most
in the
test function
of discontinuous
from characteristics),
at the interface
of degree
mechanism
schemes.
Thus
flux
]j+½ = f(j+_,u,+l), ^ u+ which
depends
both
at the interface argument continuous f(u,
a non-increasing
with respect
u) = f(u).
exact
on the left limit uj-+½ and on the right limit
xj+ ½. Here monotone
and
is then
xj+½
to each
from
following
attractive
inside
the cell Ij,
utvdx
-
by a nonlinearly
properties,
1. It can be easily determined
a scalar
high order contains
on arbitrary
then
--
Runge-Kutta
flux f(u)
the monotone
f_
1V_-
hand,
of accuracy. for efficient
1 =
solution of its first Lipschitz
in the sense that
flux is replaced the test function The final
by an v at the
semi-discrete
0
(2.5)
time discretizations
discontinuities.
multi-dimensional
triangulations,
the physical
On the other
fj+½Vj+½
strong
for any order allowing
with
function
to be at least
vj+½ and v+3-½ respectively.
+
numerical
f is a non-decreasing It is also assumed
equation,
namely
for the general
designed
the function argument.
[16] for details.
f(u)v_dx
stable
in each cell, thus
2. It can be used
than
see, e.g.
[17] may be used if the solution
u;+½ of the discontinuous
and to be consistent
rather
solver,
that
of its second
argument
Riemann
is taken
discretized
limiters
flux means
function
If (2.1) is a system
or approximate
interfaces scheme
(2.4)
The schemes
[19]. Nonlinear thus obtained
case
(1.1) with
arbitrary
In fact,
the order
of accuracy
TVB
have the
triangulations: can be locally
p adaptivity.
even those
with hanging
nodes,
thus
allowing
for efficient
h adaptivity. 3. It is extremely
local
communicate
only with
for efficient These
schemes
multi-dimensional 1. The Euler
parallel
in data
communications.
the immediate
implementations.
also have the following case
semi-discrete
(1.1) with scheme
and Crank-Nicholson,
arbitrary (2.5),
The
neighbors, See, e.g.
provable
evolution
regardless
of the solution of the order
in each
of accuracy,
cell needs thus
to
allowing
[2].
theoretical
properties,
all of these
are valid
also for the
triangulations: and
certain
have excellent
time
nonlinear
discretization stability
of it, such
properties.
as implicit
One can prove
backward a strong
L2
stabilityanda cellentropyinequalityfor the squareentropy,for the generalnonlinearscalarcase (1.1),for anyordersof accuracy on arbitrarytriangulations in anyspacedimension, withoutthe needfor nonlinearlimiters[14].Noticethat thesestabilityresultsarevalidevenwhenthesolution contains
discontinuities
2. For linear
problems
have a provable could
observe
for both 3. When
2.2. tion
error
and
nonlinear
scalar
with smooth estimate
of order
nonlinear
TVB
norm
Convection
convergence
[17, 9, 4] are
one dimensional
version
lower order
method
for the
that,
convection
the
methods
problems
The one dimensional
- (a(V)Ux)x
can (2.1),
version
be proven
stable
and stable
in the
total
in the L °c norm
of the convection
diffusion
for
equa-
= 0,
Galerkin
- (b(U)Q)x
= f_ b(U)dU.
equation
to solve
= 0,
(2.6)
method
for solving
Q - B(U)_
We can then (2.7),
resulting
(2.6) [11] approximates
formally in the
= 0,
(2.7)
use the same following
discontinuous
scheme:
find
Galerkin
u, q E V_
such
v, w E VA_,
ingredient
However,
there
In [11], criteria
for the is no longer
are given
"alternating
method
to be stable
a upwinding
for these
principle"
fluxes
in designing
_t-_-
] is chosen
all evaluated
as before
in (2.4).
at this interface
if the left value is chosen
point.
Notice The
for the former
is the
mechanism
correct
choice
of the
or characteristics
to guarantee
stability
and
numerical
to guide convergence.
fluxes
the design The best
(the
of these choice
is
the fluxes:
= B(u +) - B(u-).,
and
one
L 2 and L °¢ norms,
i
a crucial
to use the
triangulations
Ax k+l in both
k
J
j
fluxes.
for most
of degree
(1.1).
of the local discontinuous
and B(U)
J
"hats").
In effect,
polynomials
system
for all test functions
Again,
used,
problems
equations.
U, + ](U)_ b(U) = _,
piecewise
form
where a(U) >_ O. The semi-discrete
where
using
of the order
nonlinear
Vt + ](U)x
the following
methods
Ax k+½ in L 2 [15].
cases)
nonlinear
diffusion
these
problems.
limiters
for scalar
multi-dimensional
(1.2) has the
solutions,
(and prove in many
linear
variation
such as shocks.
0 = q-',
B = B(u +)
(2.9)
?.$-
_
that
we did not write
"alternating then
the subscript
principle"
the right
j +
1 for the fluxes
refers to the alternating
value is chosen
for the
latter,
as they
choices
are
of 0 and/3:
as in (2.9).
One could
also choose
0 = q+; with
all the
other
We remark in cell Ij then
fluxes
that
unchanged.
the appearance
q is eliminated
These of the
by using the
b = B(u-)
choices
of fluxes
auxiliary second
variable
equation
guarantee
stability
q is superficial: in (2.8)
and solving
and optimal when
a local
a small
linear
convergence. basis
is chosen
system
if the
localbasisis notorthogonal. Theactualscheme foru
takes
is a big advantage
"mixed
genuinely
global.
that
for the scheme The
the choice
for u after
schemes
dimensional
part
above
properties,
in the
all of these
L 2
and
(and prove
in many
smooth
cases)
convergence The same
the derivative
equations.
f(U),
r(U)
and
The semi-discrete the following
g(U) version
lower order
(2.10),
formally
resulting
both
in the
three
(1.2),
arbitrary
stencil
methods
using
of accuracy
estimate
Ux, even if q can be locally version
polynomials
a strong arbitrary diffusion
of degree
one could
eliminated
in actual
like equation
k
observe
for both
also for the auxiliary
of the KdV
+ (r'(U)g(r(U)x)x):_
on
size of the
L 2 and L _ norms,
can be obtained
backward
goes to zero.
piecewise
Ax k+l in both
Ill]:
as implicit
of the
coefficient
properties theoretical
One can prove
orders
multi
derivative
triangulations
properties.
is independent
the diffusion
first
provable
of it, such
for any
stability
of the order error
in [11].
general
attractive
Ax k in L _. In effect, for most triangulations
linear
variable
q,
calculation.
(1.3)
has the form
(2.10)
= O,
functions.
of the local discontinuous
Galerkin
method
for solving
(2.10)
[20] approximates
system
Ut+(f(U)+r'(U)P),=O, We can then
case
these
method
for the most
all of the
case (1.2) with
when
This
is usually
the most compact
also have the following
stability
The one dimensional
are arbitrary
retain
They
nonlinear
This
alone.
variable
Galerkin
yields
is nonlinear
discretization
solutions,
of order
which
time
scalar
in the limit
Ut + f(U):_ where
certain
dimension.
estimate
approximates
and
principle
(2.6), or in fact
part,
equations.
nonlinear
space
problems.
like
diffusion
auxiliary
local discontinuous
equation (1.2),
have excellent
with
error
and nonlinear
KdV
(2.8),
is also valid
problems
have a provable
equations
whose
for convection
eliminated.
for the multi-dimensional
general
in any
and hence
is termed
q is locally
derivation
scheme
for the
2. For linear
2.3.
second
Crank-Nicholson,
stability
which
diffusion
are valid
triangulations terms
variable
used on convection
semi-discrete
Euler
the scheme
to that
methods",
in (2.9) by the alternating
for the one dimensional
convection
and
that
of fluxes
the auxiliary
for the method
1. The
over the traditional
thus designed
nonlinear
convection
scheme
This is the reason
We also remark
listed
of the
a form similar
use the
in the following
same
P-g(Q)x=O, discontinuous
scheme:
find u,p,
Galerkin
Q-r(U)x=O. method
q E FAx such
that,
for the
(2.11)
convection
for all test
functions
equation
to solve
v, w, z E lgz_,,
(2.12) J
_j Again, "hats"). and
a crucial
qzdx
+ fb
ingredient
It is found
r(u)z,
dx - ,j+_l z-j+,l -}-,j_½ z;_½
for the
out in [20] that
method
to be stable
_- O.
is the
one can take the following
correct simple
choice choice
of the of fluxes
numerical to guarantee
fluxes
(the
stability
convergence:
f=f(u-,u+),
r '=r(u+)-r(u-) "L/+ -- iS--
15=p +, '
t_=_(q-,q+),
P=r(u-)
(2.13)
where
](u-,u
crucial
part
+) is a monotone is still the
flux
"alternating
]=](u-
for f(u), principle"
u+),
_,=
also
basis
work.
is chosen
(2.12)
and
takes
Again,
in cell Ij then
solving
two small
a form similar
alternating locally
to that
principle
both
of them
linear
systems
a compact
The
three
schemes
thus
designed
nonlinear
attractive provable
arbitrary
triangulations
1. The
KdV
the
Thus
_=_(q-,q+),
auxiliary
if tile local alone.
listed
theoretical
÷=r(u
+)
variables
by using
basis
q is superficial:
the
second
is not orthogonal.
We also remark
for the
p and
scheme
that
The
the choice
for u after
the
and
when
third
auxiliary
equations
actual
of fluxes
a local in
scheme
for u
in (2.13)
variables
by the
p and
q are
(1.3),
and
for any orders
the coefficients
of degree
used are
and
certain
time
linear
and
third
for the
in all the
derivatives,
for the
most
equations.
general
retain They
multi
all of the
also have
multi-dimensional
of the nonlinear
method
these
derivative with
order
discretization stability entropy,
case
the
(1.3)
with
terms
for the general
results
tend
Ax k+l in both
backward
nonlinear
scalar
dimension,
are valid
even
without
in the
limit
L2 case the
when
to zero.
solutions,
of order
as implicit
One can prove a strong
in any space
stability
smooth
of it, such
properties.
triangulations
that
these
methods
using
piecewise
Ax k+ ½ in L 2 [20]. In numerical L 2 and L _
norms
polyno-
experiments
for both
one and multiple
In this section,
cases. method
for the
for the bi-harmonic
bi-harmonic
type
equations.
type
(1.4).
We will concentrate
equation
on the
case + (a(U_)U_)x_,
= O,
0 < x < 1
(3.1)
condition g(x,O)=U°(x),
simplicity
valid
square
error estimate
Galerkin
a LDG
for the
problems
Ut + f(U)_:
and periodic
or in fact
on convection
nonlinear
on arbitrary
k have a provable
and analyze
an initial
method of these
[20]. Notice
linear
discontinuous
one dimensional
(2.10),
is nonlinear
all
inequality
of accuracy limiters
convergence
dimensional A local
equation
which
have excellent
of the dispersive
one observes
we present
(2.12),
a cell entropy
2. For one dimensional
3.
for the
properties,
scheme
need for nonlinear
mials
like
(1.3),
above
and Crank-Nicholson,
stability
KdV
[20]:
semi-discrete
Euler
for the
like equations
properties
following
The
sides.
In fact,
'
can be eliminated
stencil
flux for -g(q).
eliminated.
dimensional
with
/Z--
of the
for convection
yields
_=p-,
--
appearance
is a monotone
to take i5 and ÷ from opposite
_/+
the
-_(q-,q+)
r(u+)-r(u-)
'
would
and
boundary
conditions.
only and is not essential:
generalization
to the
multiple
0
