DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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Godunov numerical flux for Burgers' equation

Definition

For Burgers' flux $f_{B} (s) = s^{2} / 2$ , define $G ⁣ : R^{2} \to R$ as follows. If $a \leq b$ , set $G (a, b) = {\begin{cases} a^{2} / 2, & a \geq 0, \\ b^{2} / 2, & a < 0 and b \leq 0, \\ 0, & a < 0 < b . \end{cases}$ If $b < a$ , set $G (a, b) = \frac{1}{2} \max {a^{2}, b^{2}} .$ This is the Godunov flux for the convex Burgers flux: for increasing states it uses the minimum of $f_{B}$ on $[a, b]$ , and for decreasing states it uses the maximum of $f_{B}$ on $[b, a]$ . It is proposed as a particular numerical flux in the scheme of Conservative two-point finite-volume schemes for scalar laws. Its continuity and consistency are verified in the pair.

Depends on

Conservative two-point finite-volume schemes for scalar laws

Used by

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

H. De Sterck and P. Ullrich, Introduction to Computational PDEs, Chapter 3, complete chapter, PDF pp. 81–106 (standard reference, not scraped)
Benjamin Seibold, MIT OpenCourseWare 18.336, Lecture 19: Finite Volume Methods, complete six-page note (standard reference, not scraped)