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      "statement": "# Godunov: use local wave problems to calculate shocks conservatively\n\nGodunov constructs conservative gas-dynamics updates from local discontinuity-resolution problems and proves a linear monotonicity/order barrier, while the paper’s nonlinear limit and stability claims retain explicit convergence and time-step conditions.\n\nAt a shock, fluid properties jump abruptly. Treating the jump as an ordinary smooth derivative can introduce oscillations that have no physical meaning. Godunov instead treats each layer as having a constant state, resolves the elementary wave problem between neighboring layers, and uses the resulting boundary pressure and velocity to update conserved quantities. The original develops this construction in Lagrangian coordinates. It also explains a fundamental tradeoff: a linear two-time-level scheme that preserves every monotone profile cannot be second-order accurate at a noninteger Courant number. This does not forbid nonlinear high-resolution methods. For the gas calculation, conservation identifies the limit if a suitable numerical sequence converges; the paper does not prove unconditional nonlinear convergence. It reports successful shock calculations while recording entropy traces, reduced accuracy at material interfaces and conditions on time steps. The English document credits I. Bohachevsky as translator; the original scientific author is Godunov.\n\n## Scientific scope\n\nThe finite-volume scheme evolves the stated conservation laws using local Riemann problems and stability restrictions. Exact local resolution does not by itself prove convergence for every nonlinear multidimensional system or arbitrary discontinuous data. Boundary treatment, time step and thermodynamic model are part of the algorithm. Later high-order and approximate solvers remain separate.\n\n## People and systems\n\nS. K. Godunov\n\n\n\n## Original sources\n\n- [S. K. Godunov1959: completeEnglishtranslation byI.Bohachevsky,HALv1SG1959](https://hal.science/hal-01620642/file/SG1959.pdf)\n- [MathNetoriginal1959Russianpublisherbibliographicidentity/soleauthor](https://www.mathnet.ru/eng/sm4873)\n",
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