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On particle-grid interpolation and calculating chemistry in particle-in- cell methods. (English) Zbl 0807.65135

The authors consider numerical methods for model equations describing chemically reacting fluid flow. Three variants of the particle-in-cell (PIC) method are compared with a finite difference method (FDM). A 1D premixed flame problem serves as a test example. The FDM is found more accurate or more efficient than the PIC methods for fluid velocities comparable to the flame velocity. For greater fluid velocities the PIC methods are found to be superior. Further, a simplification is introduced that yields a significant increase in computational efficiency without essential increase of the error.
In Section 1 an overview over particle methods as PIC, FLIP and the discrete vortex method is given. In PIC methods a computational grid is used for the calculation of interactions between particles. In a three stage procedure particle properties are updated in time by first interpolating from particles onto the grid, then advancing the flow field on the grid, and finally, interpolating back to the particles. The accuracy of the method can strongly depend on the interpolation methods used in the first and third step. Advantages of PIC methods, and as a major motivation the ability to track material interfaces, are listed.
In Section 2 the model flame problem is formulated. Exact travelling wave solutions are given, and the structure of steady state solutions is discussed.
In Section 3 numerical formulations of three PIC methods and an FDM are presented. The interpolation methods are motivated and related to FDM and finite element approximations.
In Section 4 results are discussed. Three groups of calculations are performed, the first to assess the accuracies, the second to study variants of the treatment of the chemical source terms, and the last one to compare the methods for unsteady cases with the flame moving relatively to the grid.
In Section 5 the results are interpreted by use of a truncation error analysis.
The last section contains conclusions and a remark on open problems concerned with the range of densities that can be represented by the particles.

MSC:

65Z05 Applications to the sciences
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35Q72 Other PDE from mechanics (MSC2000)
92E20 Classical flows, reactions, etc. in chemistry
80A32 Chemically reacting flows
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