Riemann invariant

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Riemann invariants are mathematical transformations made on a system of conservation equations 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.[1]

Mathematical theory[edit]

Consider the set of conservation equations:

where and are the elements of the matrices and where and are elements of vectors. 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 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 nontrival 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 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 can be multiplied in to help integrate this. So the system now has the characteristic form

on

which is equivalent to the diagonal system[2]

The solution of this system can be given by the generalized hodograph method.[3][4]

Example[edit]

Consider the shallow water equations

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

In shallow water equations there is the relation to give the Riemann invariants

to give the equations

Which can be solved by the hodograph transformation. 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 of is not zero.

References[edit]

  1. ^ Riemann, Bernhard (1860). "Ueber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite" (PDF). Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 8. Retrieved 2012-08-08. 
  2. ^ Whitham, G. B. (1974). Linear and Nonlinear Waves. Wiley. ISBN 978-0-471-94090-6. 
  3. ^ Kamchatnov, A. M. (2000). Nonlinear Periodic Waves and their Modulations. World Scientific. ISBN 978-981-02-4407-1. 
  4. ^ Tsarev, S. P. (1985). "On Poisson brackets and one-dimensional hamiltonian systems of hydrodynamic type" (PDF). Soviet Mathematics Doklady 31 (3): 488–491. MR 87b:58030. Zbl 0605.35075.