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3. Numerics

    1. Numerics for Landau collision operator


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      Problem 3.1.

      [Jingwei Hu (Purdue University)] Among the deterministic numerical methods for the Landau equation, it is known that the Fourier spectral method can produce very accurate results and is efficient to implement. However, the spectral method doesn’t preserve the mathematical properties of the solutions such as positivity, conservation, and entropy decay. On the other hand, there exist finite difference methods that can preserve these properties yet are not accurate nor efficient to implement. The big open problem here is to search a better method that features both efficiency and accuracy and is structure preserving.

        • Numerical methods for nonlinear equation with gradient flow structure


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          Problem 3.2.

          [Li Wang (SUNY Buffalo)] Numerical methods for nonlinear equations with a gradient flow structure are expected to conserve mass, preserve positivity, dissipate the energy, and capture the steady states in the fully discrete sense. Existing methods, however, satisfy some of these desired properties, but not all. It is also preferable to have an implicit solver so that stability issue can be resolved. Therefore the open problem is to design implicit, high order, structure preserving methods.

              Cite this as: AimPL: Nonlocal differential equations in collective behavior, available at http://aimpl.org/nonlocalde.