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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. A 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.
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Scientific scope
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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People and systems
P. L. Roe
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English catalogue narrative
Roe: use a simpler wave problem while preserving the jump in conserved quantities
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.
A 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.
Scientific scope
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.
People and systems
P. L. Roe