##
**Solution of two-dimensional Riemann problems of gas dynamics by positive schemes.**
*(English)*
Zbl 0952.76060

The authors emphasize the fact that the total variation diminishing (TVD) schemes for numerically solving hyperbolic conservation laws exist only for scalar conservation laws and for linear hyperbolic systems in one space variable. TVD scheme cannot exist for non-scalar hyperbolic linear or nonlinear systems in more than one space variable. The authors further mention that the positivity principle introduced in their previous publication is adequate for multidimensional systems. This positivity principle is illustrated from
\[
U^*_J= U_J- \sum^d_{s= 1}{\Delta t\over\Delta x_s} [F_{J+1/2e_s}- F_{J- 1/2e_s}],
\]
where \(U\) stands for the dependent variables in the \(n\)-dimensional space \(x_s\), and \(F\) is the flux function of the corresponding system of unsteady equations. The subscript \(J\) is the lattice index of the numerical scheme. The positivity principle is defined when the above expression can be given by
\[
U^*_J= \sum_K C_K U_{J+K},
\]
where \(C_K\) can themselves depend on all \(U_{J+K}\), and \(C_K\) must be symmetric; each \(C_K\) is positive; also \(\sum_K C_K= I\), and \(C_K= 0\) except for a finite subset of \(K\).

There are many ways to write the form satisfying this positivity principle. In their previous publication, the authors have constructed a two-parameter family of second-order positive schemes. It was given by \[ F^{num}_{j+1/2}= {F(U_j)+ F(U_{j+1})\over 2}- {1\over 2} R[\alpha|\Lambda|(I- \Phi^0)+ \beta\text{ diag}(\mu^i)(I- \Phi^1)] R^{-1}(U_{j+ 1}- U_j). \] The second term on the right-hand side of the above expression can be given as \(R(dwf^1, dwf^2,\) \(\dots, dwf^n)^T\). \(R\) is the matrix whose columns are normalized right eigenvectors of the Jacobian matrices \(A_s\) \((A_s= \nabla F_s)\), and \(R^{-1}\) is then the matrix whose rows are normalized left eigenvectors \(\ell^k\) of \(A\). \(\Lambda\) is the diagonal matrix of eigenvalues of \(A\). Here \(\alpha\), \(\beta\) are positive constants satisfying certain conditions, and \(\Phi^0\), \(\Phi^1\) are flux limiters whose arguments and limiters themselves also satisfy certain conditions.

In the present paper the authors claim that this scheme is simple and robust. By evaluating the flux dimension-by-dimension, it is far simpler than any genuinely multidimensional approach. An additional advantage is its low cost of computations. To evaluate one flux, this scheme needs only three matrix-vector multiplications: the first is \(R^{-1}(U_{j+ 1}- U_j)\), the second is \(\ell^k(U_{j+ 1}- U_j)\) for \(k= 1,2,\dots, n\), and the third is \(R\cdot dwf\). Furthermore, the authors present a Fortran 77 computer program of their positive scheme (excluding the initial and boundary subroutines). They present results of numerical computations for different types of discontinuities along the interfaces between four quadrants with initial uniform-flow conditions. By using the symbol \(R\) for rarefaction wave, \(S\) for shock wave, and \(J\) for contact discontinuity, the authors produce results comparable with results obtained by other authors. They claim that with the fixed parameters in the first set of computations, their results are strikingly consistent with other results. In the second set of results, the slip lines are much thinner, and more vortices are seen in refined computations. The authors conclude that Glimm-type estimates in two space dimensions seem very unlikely, and also any scheme based upon genuinely multidimensional Riemann solvers would be very complicated.

There are many ways to write the form satisfying this positivity principle. In their previous publication, the authors have constructed a two-parameter family of second-order positive schemes. It was given by \[ F^{num}_{j+1/2}= {F(U_j)+ F(U_{j+1})\over 2}- {1\over 2} R[\alpha|\Lambda|(I- \Phi^0)+ \beta\text{ diag}(\mu^i)(I- \Phi^1)] R^{-1}(U_{j+ 1}- U_j). \] The second term on the right-hand side of the above expression can be given as \(R(dwf^1, dwf^2,\) \(\dots, dwf^n)^T\). \(R\) is the matrix whose columns are normalized right eigenvectors of the Jacobian matrices \(A_s\) \((A_s= \nabla F_s)\), and \(R^{-1}\) is then the matrix whose rows are normalized left eigenvectors \(\ell^k\) of \(A\). \(\Lambda\) is the diagonal matrix of eigenvalues of \(A\). Here \(\alpha\), \(\beta\) are positive constants satisfying certain conditions, and \(\Phi^0\), \(\Phi^1\) are flux limiters whose arguments and limiters themselves also satisfy certain conditions.

In the present paper the authors claim that this scheme is simple and robust. By evaluating the flux dimension-by-dimension, it is far simpler than any genuinely multidimensional approach. An additional advantage is its low cost of computations. To evaluate one flux, this scheme needs only three matrix-vector multiplications: the first is \(R^{-1}(U_{j+ 1}- U_j)\), the second is \(\ell^k(U_{j+ 1}- U_j)\) for \(k= 1,2,\dots, n\), and the third is \(R\cdot dwf\). Furthermore, the authors present a Fortran 77 computer program of their positive scheme (excluding the initial and boundary subroutines). They present results of numerical computations for different types of discontinuities along the interfaces between four quadrants with initial uniform-flow conditions. By using the symbol \(R\) for rarefaction wave, \(S\) for shock wave, and \(J\) for contact discontinuity, the authors produce results comparable with results obtained by other authors. They claim that with the fixed parameters in the first set of computations, their results are strikingly consistent with other results. In the second set of results, the slip lines are much thinner, and more vortices are seen in refined computations. The authors conclude that Glimm-type estimates in two space dimensions seem very unlikely, and also any scheme based upon genuinely multidimensional Riemann solvers would be very complicated.

Reviewer: When L.Chow (Boca Raton)

### MSC:

76M20 | Finite difference methods applied to problems in fluid mechanics |

76L05 | Shock waves and blast waves in fluid mechanics |

76N15 | Gas dynamics (general theory) |

### Keywords:

two-dimensional Riemann problem; total variation diminishing schemes; positivity principle; multidimensional systems; flux limiters; matrix-vector multiplications; Fortran 77 computer program; positive scheme; rarefaction wave; shock wave; contact discontinuity
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\textit{P. D. Lax} and \textit{X.-D. Liu}, SIAM J. Sci. Comput. 19, No. 2, 319--340 (1998; Zbl 0952.76060)

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### References:

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