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 :
l i ( A i j ∂ u j ∂ t + a i j ∂ u j ∂ x ) + l j b j = 0 {\displaystyle l_{i}\left(A_{ij}{\frac {\partial u_{j}}{\partial t}}+a_{ij}{\frac {\partial u_{j}}{\partial x}}\right)+l_{j}b_{j}=0} where A i j {\displaystyle A_{ij}} and a i j {\displaystyle a_{ij}} are the elements of the matrices A {\displaystyle \mathbf {A} } and a {\displaystyle \mathbf {a} } where l i {\displaystyle l_{i}} and b i {\displaystyle b_{i}} are elements of vectors . It will be asked if it is possible to rewrite this equation to
m j ( β ∂ u j ∂ t + α ∂ u j ∂ x ) + l j b j = 0 {\displaystyle m_{j}\left(\beta {\frac {\partial u_{j}}{\partial t}}+\alpha {\frac {\partial u_{j}}{\partial x}}\right)+l_{j}b_{j}=0} To do this curves will be introduced in the ( x , t ) {\displaystyle (x,t)} plane defined by the vector field ( α , β ) {\displaystyle (\alpha ,\beta )} . The term in the brackets will be rewritten in terms of a total derivative where x , t {\displaystyle x,t} are parametrized as x = X ( η ) , t = T ( η ) {\displaystyle x=X(\eta ),t=T(\eta )}
d u j d η = T ′ ∂ u j ∂ t + X ′ ∂ u j ∂ x {\displaystyle {\frac {du_{j}}{d\eta }}=T'{\frac {\partial u_{j}}{\partial t}}+X'{\frac {\partial u_{j}}{\partial x}}} comparing the last two equations we find
α = X ′ ( η ) , β = T ′ ( η ) {\displaystyle \alpha =X'(\eta ),\beta =T'(\eta )} which can be now written in characteristic form
m j d u j d η + l j b j = 0 {\displaystyle m_{j}{\frac {du_{j}}{d\eta }}+l_{j}b_{j}=0} where we must have the conditions
l i A i j = m j T ′ {\displaystyle l_{i}A_{ij}=m_{j}T'} l i a i j = m j X ′ {\displaystyle l_{i}a_{ij}=m_{j}X'} where m j {\displaystyle m_{j}} can be eliminated to give the necessary condition
l i ( A i j X ′ − a i j T ′ ) = 0 {\displaystyle l_{i}(A_{ij}X'-a_{ij}T')=0} so for a nontrivial solution is the determinant
| A i j X ′ − a i j T ′ | = 0 {\displaystyle |A_{ij}X'-a_{ij}T'|=0} For Riemann invariants we are concerned with the case when the matrix A {\displaystyle \mathbf {A} } is an identity matrix to form
∂ u i ∂ t + a i j ∂ u j ∂ x = 0 {\displaystyle {\frac {\partial u_{i}}{\partial t}}+a_{ij}{\frac {\partial u_{j}}{\partial x}}=0} notice this is homogeneous due to the vector n {\displaystyle \mathbf {n} } being zero. In characteristic form the system is
l i d u i d t = 0 {\displaystyle l_{i}{\frac {du_{i}}{dt}}=0} with d x d t = λ {\displaystyle {\frac {dx}{dt}}=\lambda } Where l {\displaystyle l} is the left eigenvector of the matrix A {\displaystyle \mathbf {A} } and λ ′ s {\displaystyle \lambda 's} is the characteristic speeds of the eigenvalues of the matrix A {\displaystyle \mathbf {A} } which satisfy
| A − λ δ i j | = 0 {\displaystyle |A-\lambda \delta _{ij}|=0} To simplify these characteristic equations we can make the transformations such that d r d t = l i d u i d t {\displaystyle {\frac {dr}{dt}}=l_{i}{\frac {du_{i}}{dt}}}
which form
μ l i d u i = d r {\displaystyle \mu l_{i}du_{i}=dr} An integrating factor μ {\displaystyle \mu } can be multiplied in to help integrate this. So the system now has the characteristic form
d r d t = 0 {\displaystyle {\frac {dr}{dt}}=0} on d x d t = λ i {\displaystyle {\frac {dx}{dt}}=\lambda _{i}} which is equivalent to the diagonal system [ 2]
r t k + λ k r x k = 0 , {\displaystyle r_{t}^{k}+\lambda _{k}r_{x}^{k}=0,} k = 1 , . . . , N . {\displaystyle k=1,...,N.} The solution of this system can be given by the generalized hodograph method .[ 3] [ 4]
Consider the one-dimensional Euler equations written in terms of density ρ {\displaystyle \rho } and velocity u {\displaystyle u} are
ρ t + ρ u x + u ρ x = 0 {\displaystyle \rho _{t}+\rho u_{x}+u\rho _{x}=0} u t + u u x + ( c 2 / ρ ) ρ x = 0 {\displaystyle u_{t}+uu_{x}+(c^{2}/\rho )\rho _{x}=0} with c {\displaystyle c} being the speed of sound is introduced on account of isentropic assumption. Write this system in matrix form
( ρ u ) t + ( u ρ c 2 ρ u ) ( ρ u ) x = ( 0 0 ) {\displaystyle \left({\begin{matrix}\rho \\u\end{matrix}}\right)_{t}+\left({\begin{matrix}u&\rho \\{\frac {c^{2}}{\rho }}&u\end{matrix}}\right)\left({\begin{matrix}\rho \\u\end{matrix}}\right)_{x}=\left({\begin{matrix}0\\0\end{matrix}}\right)} where the matrix a {\displaystyle \mathbf {a} } from the analysis above the eigenvalues and eigenvectors need to be found. The eigenvalues are found to satisfy
λ 2 − 2 u λ + u 2 − c 2 = 0 {\displaystyle \lambda ^{2}-2u\lambda +u^{2}-c^{2}=0} to give
λ = u ± c {\displaystyle \lambda =u\pm c} and the eigenvectors are found to be
( 1 c ρ ) , ( 1 − c ρ ) {\displaystyle \left({\begin{matrix}1\\{\frac {c}{\rho }}\end{matrix}}\right),\left({\begin{matrix}1\\-{\frac {c}{\rho }}\end{matrix}}\right)} where the Riemann invariants are
r 1 = J + = u + ∫ c ρ d ρ , {\displaystyle r_{1}=J_{+}=u+\int {\frac {c}{\rho }}d\rho ,} r 2 = J − = u − ∫ c ρ d ρ , {\displaystyle r_{2}=J_{-}=u-\int {\frac {c}{\rho }}d\rho ,} ( J + {\displaystyle J_{+}} and J − {\displaystyle J_{-}} are the widely used notations in gas dynamics ). For perfect gas with constant specific heats, there is the relation c 2 = const γ ρ γ − 1 {\displaystyle c^{2}={\text{const}}\,\gamma \rho ^{\gamma -1}} , where γ {\displaystyle \gamma } is the specific heat ratio , to give the Riemann invariants[ 5] [ 6]
J + = u + 2 γ − 1 c , {\displaystyle J_{+}=u+{\frac {2}{\gamma -1}}c,} J − = u − 2 γ − 1 c , {\displaystyle J_{-}=u-{\frac {2}{\gamma -1}}c,} to give the equations
∂ J + ∂ t + ( u + c ) ∂ J + ∂ x = 0 {\displaystyle {\frac {\partial J_{+}}{\partial t}}+(u+c){\frac {\partial J_{+}}{\partial x}}=0} ∂ J − ∂ t + ( u − c ) ∂ J − ∂ x = 0 {\displaystyle {\frac {\partial J_{-}}{\partial t}}+(u-c){\frac {\partial J_{-}}{\partial x}}=0} In other words,
d J + = 0 , J + = const along C + : d x d t = u + c , d J − = 0 , J − = const along C − : d x d t = u − c , {\displaystyle {\begin{aligned}&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{aligned}}} where C + {\displaystyle C_{+}} and C − {\displaystyle 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
A ∂ v ∂ t + B ∂ v ∂ x = 0 {\displaystyle A{\frac {\partial v}{\partial t}}+B{\frac {\partial v}{\partial x}}=0} Then it may be possible to multiply across by the inverse matrix A − 1 {\displaystyle A^{-1}} so long as the matrix determinant of A {\displaystyle \mathbf {A} } is not zero.
^ 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 . ^ Whitham, G. B. (1974). Linear and Nonlinear Waves . Wiley . ISBN 978-0-471-94090-6 . ^ Kamchatnov, A. M. (2000). Nonlinear Periodic Waves and their Modulations . World Scientific . ISBN 978-981-02-4407-1 . ^ Tsarev, S. P. (1985). "On Poisson brackets and one-dimensional hamiltonian systems of hydrodynamic type" (PDF) . Soviet Mathematics - Doklady . 31 (3): 488–491. MR 2379468 . Zbl 0605.35075 . Archived from the original (PDF) on 2012-03-30. Retrieved 2011-08-20 . ^ Zelʹdovich, I. B., & Raĭzer, 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.