词条 | Riemann invariant |
释义 |
Mathematical theoryConsider 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] ExampleConsider the one-dimensional Euler equations written in terms of density and velocity are with being the speed of sound is introduced on account of isentropic assumption. 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 ( and are the widely used notations in gas dynamics). For perfect gas with constant specific heats, there is the relation , where is the specific heat ratio, to give the Riemann invariants[5][6] to give the equations In other words, where and 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 so long as the matrix determinant of is not zero. See also
References1. ^{{cite journal |last1=Riemann |first1=Bernhard |year=1860 |title=Ueber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite |journal=Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen |volume=8 |url=http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Welle/Welle.pdf |accessdate=2012-08-08 }} 2. ^{{cite book |last=Whitham |first=G. B. |year=1974 |title=Linear and Nonlinear Waves |publisher=Wiley |page= |isbn=978-0-471-94090-6}} 3. ^{{cite book |last=Kamchatnov |first=A. M. |year=2000 |title=Nonlinear Periodic Waves and their Modulations |publisher=World Scientific |page= |isbn=978-981-02-4407-1}} 4. ^{{cite journal |last=Tsarev |first=S. P. |year=1985 |title=On Poisson brackets and one-dimensional hamiltonian systems of hydrodynamic type |url=http://ikit.institute.sfu-kras.ru/files/ikit/Tsarev-DAN-1985-eng.pdf |journal=Soviet Mathematics - Doklady |volume=31 |issue=3 |pages=488–491 |mr=2379468 |zbl=0605.35075}} 5. ^Zelʹdovich, I. B., & Raĭzer, I. P. (1966). Physics of shock waves and high-temperature hydrodynamic phenomena (Vol. 1). Academic Press. 6. ^Courant, R., & Friedrichs, K. O. 1948 Supersonic flow and shock waves. New York: Interscience. 4 : Invariant theory|Partial differential equations|Conservation equations|Bernhard Riemann |
随便看 |
|
开放百科全书收录14589846条英语、德语、日语等多语种百科知识,基本涵盖了大多数领域的百科知识,是一部内容自由、开放的电子版国际百科全书。