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.
Mathematical theory 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
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