An investigation of discrete adjoints for flux-limited numerical schemes

ABSTRACT

In this paper we investigate the construction of discrete adjoints of flux-limited numerical schemes for the advection equation. Discrete adjoints are attractive since they can be generated easily via automatic differentiation. However, a careful analysis is needed in order to understand their properties. We discuss several issues posed by the differentiation of the limiter functions and propose an alternative implementation of the forward model that leads to stable discrete adjoint solutions.

REFERENCES

• 1: R. Giering. Tangent linear and Adjoint Model Compiler User's Manual 1.4, 1999.
• 2: L. Hascoët and V. Pascual. TAPENADE 2.1 User's Guide. Technical Report 0300, INRIA, 2004.
• 3: W. Hundsdorfer , B. Koren , M. van Loon , J. G. Verwer, A positive finite-difference advection scheme, Journal of Computational Physics, v.117 n.1, p.35-46, March 1, 1995 [doi>10.1006/jcph.1995.1042]
• 4: R. J. Leveque. Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, 1st edition, 2002.
• 5: Z. Liu and A. Sandu. Analysis of discrete adjoints for upwind numerical schemes, volume 3515 of Lecture Notes in Computer Science, pages 829--836. Springer Berlin / Heidelberg, 2005.
• 6: Z. Sirkes and E. Tziperman. Finite difference of adjoint or adjoint of finite difference? Monthly Weather Review, 125(12):3373--3378, 1997.
• 7: B. van Leer. Towards the ultimate conservative difference scheme: IV. A new approach to numerical convection. Journal of Computational Physics, 23:276--299, 1977.

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.6 SIMULATION AND MODELING
      I.6.5 Model Development
          Subjects: Modeling methodologies

Keywords:
advection equation, discrete adjoint model, flux limiter