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        "description": "The approximate solver uses a state-dependent linearization satisfying the stated consistency, conservation and wave-decomposition properties. Degeneracy, physical admissibility and entropy behavior need care; an entropy fix may be necessary. It is not a universal exact Riemann solver or automatic positivity/convergence guarantee for every conservation law.",
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      "statement": "# Roe: use a simpler wave problem while preserving the jump in conserved quantities\n\nRoe constructed approximate Riemann solvers with consistent flux-jump linearizations and a parameter-vector representation simplifying perfect-gas Euler equations, while explicitly identifying entropy and multidimensional limitations.\n\nA fluid calculation divides space into cells. At each boundary, neighbouring states may disagree sharply; an exact wave solution can be expensive and provide more detail than the surrounding numerical scheme can use. Roe replaces this local nonlinear problem with a linear wave system chosen to preserve the full flux jump, approach the physical Jacobian for nearby states and have independent wave directions. A parameter vector makes the perfect-gas state and flux quantities quadratic, simplifying the matrix and its waves. This can recognize isolated discontinuities and provide useful information directly to a difference scheme. The1981 paper compares one shock-tube calculation on historical hardware, but explicitly warns that the base approximation can leave an unphysical expansion discontinuity intact. Additional entropy treatment is needed in such cases; positivity and general multidimensional convergence are also not consequences of the flux-jump property alone.\n\n## Scientific scope\n\nThe approximate solver uses a state-dependent linearization satisfying the stated consistency, conservation and wave-decomposition properties. Degeneracy, physical admissibility and entropy behavior need care; an entropy fix may be necessary. It is not a universal exact Riemann solver or automatic positivity/convergence guarantee for every conservation law.\n\n## People and systems\n\nP. L. Roe\n\n\n\n## Original sources\n\n- [P.L.Roe, Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes(1981),357-372 — original article web text](https://www.researchgate.net/profile/Philip-Roe-2/publication/222453065_Approximate_Riemann_Solvers_Parameter_Vector_and_Difference_Schemes/links/5c83fb6b92851c695067e90c/Approximate-Riemann-Solvers-Parameter-Vector-and-Difference-Schemes.pdf)\n",
      "summary": "Roe constructed approximate Riemann solvers with consistent flux-jump linearizations and a parameter-vector representation simplifying perfect-gas Euler equations, while explicitly identifying entropy and multidimensional limitations.\n\nA fluid calculation divides space into cells. At each boundary, neighbouring states may disagree sharply; an exact wave solution can be expensive and provide more detail than the surrounding numerical scheme can use. Roe replaces this local nonlinear problem with a linear wave system chosen to preserve the full flux jump, approach the physical Jacobian for nearby states and have independent wave directions. A parameter vector makes the perfect-gas state and flux quantities quadratic, simplifying the matrix and its waves. This can recognize isolated discontinuities and provide useful information directly to a difference scheme. The1981 paper compares one shock-tube calculation on historical hardware, but explicitly warns that the base approximation can leave an unphysical expansion discontinuity intact. Additional entropy treatment is needed in such cases; positivity and general multidimensional convergence are also not consequences of the flux-jump property alone.",
      "title": "Roe: use a simpler wave problem while preserving the jump in conserved quantities"
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