Riemann invariants are
mathematical transformations made on a system of
conservation equation
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, co ...
s to make them more easily solvable. Riemann invariants are constant along the
characteristic curves of the partial differential equations where they obtain the name
invariant. They were first obtained by
Bernhard Riemann in his work on plane waves in gas dynamics.
Mathematical theory
Consider the set of
conservation equations:
:
where
and
are the
elements of the
matrices
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and
where
and
are elements of
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
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Vector may also refer to:
Mathematic ...
s. It will be asked if it is possible to rewrite this equation to
:
To do this curves will be introduced in the
plane defined by the
vector field . The term in the brackets will be rewritten in terms of a
total derivative
In mathematics, the total derivative of a function at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with res ...
where
are parametrized as
:
comparing the last two equations we find
:
which can be now written in
characteristic form
:
where we must have the conditions
:
:
where
can be eliminated to give the necessary condition
:
so for a
nontrivial solution is the determinant
:
For Riemann invariants we are concerned with the case when the matrix
is an
identity matrix to form
:
notice this is
homogeneous due to the vector
being zero. In characteristic form the system is
:
with
Where
is the left
eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
of the matrix
and
is the
characteristic speeds of the
eigenvalues of the matrix
which satisfy
:
To simplify these
characteristic equations we can make the transformations such that
which form
:
An
integrating factor
In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calcul ...
can be multiplied in to help integrate this. So the system now has the characteristic form
:
on
which is equivalent to the
diagonal system
:
The solution of this system can be given by the generalized
hodograph method.
[
]
Example
Consider the one-dimensional
Euler equations
200px, Leonhard Euler (1707–1783)
In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include ...
written in terms of density
and velocity
are
:
:
with
being the
speed of sound is introduced on account of isentropic assumption. Write this system in matrix form
:
where the matrix
from the analysis above the eigenvalues and eigenvectors need to be found. The eigenvalues are found to satisfy
:
to give
:
and the eigenvectors are found to be
:
where the Riemann invariants are
:
:
(
and
are the widely used notations in
gas dynamics
Compressible flow (or gas dynamics) is the branch of fluid mechanics that deals with flows having significant changes in fluid density. While all flows are compressible, flows are usually treated as being incompressible when the Mach number (the ...
). For perfect gas with constant specific heats, there is the relation
, where
is the
specific heat ratio, to give the Riemann invariants
[Courant, R., & Friedrichs, K. O. 1948 Supersonic flow and shock waves. New York: Interscience.]
:
:
to give the equations
:
:
In other words,
:
where
and
are the characteristic curves. This can be solved by the
hodograph transformation. In the hodographic plane, if all the characteristics collapses into a single curve, then we obtain
simple waves. If the matrix form of the system of pde's is in the form
:
Then it may be possible to multiply across by the inverse matrix
so long as the matrix
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of
is not zero.
See also
*
Simple wave
References
{{Bernhard Riemann
Invariant theory
Partial differential equations
Conservation equations
Bernhard Riemann