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A Sequel of Inverse Lax–Wendroff High Order Wall Boundary Treatment for Conservation Laws

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Abstract

When solving CFD problems, the solver, or the numerical code, plays an important role. Depending on the phenomena and problem domain, designing such numerical codes can be hard work. One strategy is to start with simple problems and construct the code as building blocks. The purpose of this work is to provide a detailed review of the theory to compute analytical and exact solutions, and recent numerical methods to construct a code to solve compressible and inviscid fluid flows with high-resolution, arbitrary domains, non-linear phenomena, and on rectangular meshes. We also propose a modification to the inverse Lax–Wendroff procedure solid wall treatment and two-dimensional WENO-type extrapolation stencil selection and weights to handle more generic situations. To test our modifications, we use the finite difference method, Lax–Friedrichs splitting, WENO-Z+ scheme, and third-order strong stability preserving Runge-Kutta time discretization. Our first problem is a simple one-dimensional transient problem with periodic boundary conditions, which is useful for constructing the core solver. Then, we move to the one-dimensional Rayleigh flow, which can handle flows with heat exchange and requires more detailed boundary treatment. The next problem is the quasi-one-dimensional nozzle flow with and without shock, where the boundary treatment needs a few adjustments. The first two-dimensional problem is the Ringleb flow, and despite being smooth, it has a curved wall as the left boundary. Finally, the last problem is a two-dimensional conical flow, which presents an oblique shock and an inclined straight line wall being the cone surface. We show that the designed accuracy is being reached for smooth problems, that high-resolution is being attained for non-smooth problems, and that our modifications produce similar results while providing a more generic way to treat solid walls.

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Funding

Funding was provided by National Science Foundation (Grant. No. DMS-1719410).

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Authors and Affiliations

  1. Mathematics and Statistics Institute, Rio de Janeiro State University, Rio de Janeiro, Brazil

    Rafael B. de Rezende Borges

  2. Laboratory of Numerical Experimentation, Department of Mechanical Engineering, Federal University of Paraná, Curitiba, Brazil

    Nicholas Dicati P. da Silva

  3. Department of Mechanical Engineering, Federal University of Technology - Paraná, Pato Branco, Brazil

    Francisco A. A. Gomes

  4. Division of Applied Mathematics, Brown University, Providence, USA

    Chi-Wang Shu & Sirui Tan

Authors
  1. Rafael B. de Rezende Borges
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  2. Nicholas Dicati P. da Silva
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  3. Francisco A. A. Gomes
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  4. Chi-Wang Shu
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  5. Sirui Tan
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Correspondence to Nicholas Dicati P. da Silva.

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The research of C.-W. Shu is supported by NSF Grant DMS-1719410.

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Borges, R.B.d.R., da Silva, N.D.P., Gomes, F.A.A. et al. A Sequel of Inverse Lax–Wendroff High Order Wall Boundary Treatment for Conservation Laws. Arch Computat Methods Eng 28, 2315–2329 (2021). https://doi.org/10.1007/s11831-020-09454-w

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  • DOI: https://doi.org/10.1007/s11831-020-09454-w

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