Riemann invariant

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Riemann invariants are mathematical transformations made on a system of quasi-linear first order partial differential 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 hyperbolic partial differential equations of the form

 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_{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

  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) 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)

 \frac{d u_j}{d \eta}=T'\frac{\partial u_j}{\partial t}+X'\frac{\partial u_j}{\partial x}

comparing the last two equations we find

 \alpha=X'(\eta), \beta=T'(\eta)

which can be now written in characteristic form

 m_j\frac{du_j  }{ d \eta  }+l_jb_j = 0

where we must have the conditions

l_iA_{ij}=m_jT'
l_ia_{ij}=m_jX'

where m_j can be eliminated to give the necessary condition

l_i(A_{ij}X'-a_{ij}T')=0

so for a nontrival solution is the determinant

|A_{ij}X'-a_{ij}T'|=0

For Riemann invariants we are concerned with the case when the matrix \mathbf{A} is an identity matrix to form

 \frac{\partial u_i}{\partial t}  +a_{ij}\frac{\partial u_j}{\partial x}=0

notice this is homogeneous due to the vector \mathbf{n} being zero. In characteristic form the system is

 l_i\frac{du_i  }{dt  }=0 with \frac{dx }{dt  }=\lambda

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

 |A -\lambda\delta_{ij}|=0

To simplify these characteristic equations we can make the transformations such that  \frac{dr_i}{dt}=l_i\frac{du_i}{dt}

which form

 \mu l_idu_i =dr_i

An integrating factor \mu can be multiplied in to help integrate this. So the system now has the characteristic form

 \frac{dr_i }{dt }=0 on \frac{dx}{dt}=\lambda_i

which is equivalent to the diagonal system[2]

r_t^k +\lambda_kr_x^k=0, k=1,...,N.

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

Example[edit]

Consider the shallow water equations

 \rho_t+\rho u_x+u\rho_x=0
 u_t+uu_x+(c^2/\rho)\rho_x=0

write this system in matrix form

 \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 \mathbf{a} from the analysis above the eigenvalues and eigenvectors need to be found.The eigenvalues are found to satisfy

  \lambda^2-2u\lambda+u^2-c^2=0

to give

 \lambda=u\pm c

and the eigenvectors are found to be

 \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=u+\int \frac{c}{\rho}d\rho,
 r_2=u-\int \frac{c}{\rho}d\rho,

In shallow water equations there is the relation c=\sqrt{\rho} to give the riemann invariants

 r_1=u+2\sqrt{\rho},
 r_2=u-2\sqrt{\rho},

to give the equations

 \frac{\partial r_1}{\partial t}+(u+\sqrt{\rho})\frac{\partial r_1}{\partial x}=0
 \frac{\partial r_2}{\partial t}+(u-\sqrt{\rho})\frac{\partial r_2}{\partial x}=0

Which can be solved by the hodograph transformation. If the matrix form of the system of pde's is in the form

 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} so long as the matrix determinant of \mathbf{A} is not zero.

References[edit]

  1. ^ Riemann, Bernhard (1860). "Ueber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite". 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". Soviet Mathematics Doklady 31 (3): 488–491. MR 87b:58030. Zbl 0605.35075.