What is recorded here?
Lax and Wendroff identify the weak limit of a boundedly convergent conservative difference scheme and construct higher-accuracy difference formulas with artificial-viscosity treatments for shock calculations. A shock is a sudden jump, so the derivatives in a fluid equation can cease to make ordinary sense. Conservation can still be expressed by integrating over space and time. Lax and Wendroff show how a discrete flux shared by neighbouring cells preserves this structure: under the required consistency and regularity, a boundedly convergent numerical sequence has a weak limit satisfying the conservation law. This identifies the limit; it does not supply convergence or physical entropy admissibility by itself. The report then develops a second-order construction for smooth parts of the solution and additional viscosity terms near shocks. Its stability analysis uses localized linearized tests and stated assumptions; some shock-width and system arguments are approximate or conjectural. Two hydrodynamic calculations illustrate the method. Both the theoretical distinction and the numerical construction became important tools, while later nonoscillatory, entropy and convergence results retain their own authorship.
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Scientific scope
The conservation argument requires consistency and convergence of the stated conservative approximations; it does not prove that every such scheme converges. Entropy/admissibility and multidimensional/system behavior need further conditions. The selected report precedes the later published version, and later high-resolution schemes receive separate credit.
Read the original work
- PeterLax and BurtonWendroff, Systems of Conservation Laws, LA2285 — original1959distributed report
- PhilippBirken and ViktorLinders, Conservation Properties of Iterative Methods for Implicit Discretizations of Conservation Laws(2022) — selected regularity definition only
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People and systems
Peter Lax; Burton Wendroff
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English catalogue narrative
Lax and Wendroff: preserve conservation when approximating waves and shocks
Lax and Wendroff identify the weak limit of a boundedly convergent conservative difference scheme and construct higher-accuracy difference formulas with artificial-viscosity treatments for shock calculations.
A shock is a sudden jump, so the derivatives in a fluid equation can cease to make ordinary sense. Conservation can still be expressed by integrating over space and time. Lax and Wendroff show how a discrete flux shared by neighbouring cells preserves this structure: under the required consistency and regularity, a boundedly convergent numerical sequence has a weak limit satisfying the conservation law. This identifies the limit; it does not supply convergence or physical entropy admissibility by itself. The report then develops a second-order construction for smooth parts of the solution and additional viscosity terms near shocks. Its stability analysis uses localized linearized tests and stated assumptions; some shock-width and system arguments are approximate or conjectural. Two hydrodynamic calculations illustrate the method. Both the theoretical distinction and the numerical construction became important tools, while later nonoscillatory, entropy and convergence results retain their own authorship.
Scientific scope
The conservation argument requires consistency and convergence of the stated conservative approximations; it does not prove that every such scheme converges. Entropy/admissibility and multidimensional/system behavior need further conditions. The selected report precedes the later published version, and later high-resolution schemes receive separate credit.
People and systems
Peter Lax; Burton Wendroff