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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.JOURNAL, Riemann, Bernhard, 1860, Ueber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite, Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 8,www.maths.tcd.ie/pub/HistMath/People/Riemann/Welle/Welle.pdf, 2012-08-08,

Mathematical theory

Consider the set of conservation equations: where A_{ij} and a_{ij} are the elements of the matrices mathbf{A} and mathbf{a} where l_{i} and b_{i} 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 (x,t) plane defined by the vector field (alpha,beta). The term in the brackets will be rewritten in terms of a total derivative where x,t are parametrized as x=X(eta),t=T(eta) comparing the last two equations we find which can be now written in characteristic form where we must have the conditions where m_j 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 mathbf{A} is an identity matrix to form notice this is homogeneous due to the vector mathbf{n} being zero. In characteristic form the system is Where l is the left eigenvector of the matrix mathbf{A} and lambda ‘s is the characteristic speeds of the eigenvalues of the matrix mathbf{A} which satisfy To simplify these characteristic equations we can make the transformations such that frac{dr}{dt}=l_ifrac{du_i}{dt}which form An integrating factor mu can be multiplied in to help integrate this. So the system now has the characteristic form which is equivalent to the diagonal systemBOOK , The solution of this system can be given by the generalized hodograph method.BOOK , JOURNAL

Example

Consider the one-dimensional Euler equations written in terms of density rho and velocity u are with c being the speed of sound is introduced on account of isentropic assumption. Write this system in matrix form where the matrix mathbf{a} 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 (J_+ and J_- are the widely used notations in gas dynamics). For perfect gas with constant specific heats, there is the relation c^2=text{const}, gamma rho^{gamma-1}, where gamma is the specific heat ratio, to give the Riemann invariantsZelʹdovich, I. B., & RaÄ­zer, I. P. (1966). Physics of shock waves and high-temperature hydrodynamic phenomena (Vol. 1). Academic Press.Courant, R., & Friedrichs, K. O. 1948 Supersonic flow and shock waves. New York: Interscience. to give the equations In other words, begin{align}&dJ_+ = 0, , J_+=text{const}quad text{along},, C_+, :, frac{dx}{dt}=u+c, &dJ_- = 0, , J_-=text{const}quad text{along},, C_-, :, frac{dx}{dt}=u-c,end{align}where C_+ and C_- 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 A^{-1} so long as the matrix determinant of mathbf{A} is not zero.

See also

• Simple wave

References

{{reflist}}{{Bernhard Riemann}}