Abstract
Source terms often arise in Computational Fluid Dynamics to describe a variety of physical phenomena, including turbulence, chemical reactions, and certain methods used for code verification, such as the method of manufactured solutions. While much has already been published on the treatment of source terms, here we follow an uncommon approach, designing compatible source term discretizations in terms of spatial truncation error for finite volume schemes in multiple dimensions. In this work we examine the effect of source term discretization on three finite volume flux schemes applied to steady flows: constant reconstruction, linear reconstruction, and a recently published third-order flux correction method. Three source term discretization schemes are considered, referred to as point, Galerkin, and corrected. The corrected source discretization is a new method that extends our previous work on flux correction to equations with source terms. In all cases, computational grid refinement studies confirm the compatibility (or lack thereof) of various flux-source combinations predicted through detailed truncation error analysis.
Similar content being viewed by others
References
LeVeque, R.: Numerical Methods for Conservation Laws. Birkhauser, Basel (1990)
Godlewski, E., Raviart, P.A.: Numerical Approximation of Hyperbolic Systems of Conservation Laws, of Applied Mathematical Sciences. Springer, New York (1996)
Toro, E.: Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Approach. Springer, New York (1997)
Murillo, J., Garcia-Navarro, P.: Weak solutions for partial differential equations with source terms: application to the shallow water equations. J. Comput. Phys. 229, 4327–4368 (2010)
Dumbser, M., Enaux, C., Toro, E.: Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. J. Comput. Phys. 227(8), 3971–4001 (2008)
Roache, P.: Verification of codes and calculations. J. Comput. Phys. 36(0001–1452), 696–702 (1998)
Roache, P.: Code verification by the method of manufactured solutions. J. Comput. Phys. 124, 4–10 (2002)
Lomax, H., Pulliam, T.H., Zingg, D.W.: Fundamentals of Computational Fluid Dynamics. Springer, New York (2001)
Oberkampf, W., Roy, C.: Verification and Validation in Scientific Computing. Cambridge University Press, Cambridge (2010)
Katz, A., Sankaran, V.: An efficient correction method to obtain a formally third-order accurate flow solver for node-centered unstructured grids. J. Sci. Comput. 51, 375–393 (2012)
Boris, J.P., Book, D.L.: Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works. J. Comput. Phys. 11(1), 38–69 (1973)
Salari, K., Knupp, P.: Code verification by the method of manufactured solutions. Technical Report SAND2000-1444, Sandia National Laboratories (2000)
Roy, C.J.: Review of code and solution verification procedures for computational simulation. J. Comput. Phys. 205, 131–156 (2005)
Eca, L., Hoekstra, M., Hay, A., Pelletier, D.: A manufactured solution for a two-dimensional steady wall-bounded incompressible turbulent flow. J. Comput. Phys. 21, 175–188 (2007)
Folkner, D., Katz, A., Sankaran, V.: Design and verification methodology of boundary conditions for finite volume schemes. J. Comput. Phys. 96, 264–275 (2014)
Katz, A., Sankaran, V.: Mesh quality effects on the accuracy of Euler and Navier-Stokes solutions on unstructured meshes. J. Comput. Phys. 230(20), 7670–7686 (2011)
Katz, A., Sankaran, V.: High aspect ratio grid effects on the accuracy of Navier-Stokes solutions on unstructured meshes. J. Comput. Phys. 65, 67–79 (2012)
Work, D., Tong, O., Workman, R., Katz, A., Wissink, A.: Strand grid solution procedures for sharp corners. J. Comput. Phys. 52(7), 1528–1541 (2014)
Jameson, A.: Advances in bringing high-order methods to practical applications in computational fluid dynamics. AIAA paper 2011–3226, AIAA 20th Computational Fluid Dynamics Conference, Honolulu, HI, (2011)
Toro, E., Titarev, V.: ADER schemes for scalar hyperbolic conservation laws with source terms in three space dimensions. J. Comput. Phys. 202(1), 196–215 (2005)
Toro, E., Titarev, V.: Derivative Riemann solvers for systems of conservation laws and ader methods. J. Comput. Phys. 212(1), 150–165 (2006)
Vazquez-Cendon, M.: Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry. J. Comput. Phys. 148, 497–526 (1999)
Bermudez, A., Vazquez, M.E.: Upwind methods for hyperbolic conservation laws with source terms. J. Comput. Phys. 23(8), 1049 (1994)
Bermudez, A., Dervieux, A., Desideri, J.A., Vazquez, M.E.: Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes. Comput. Methods Appl. Mech. Eng. 155(49), 49–72 (1998)
Leveque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, New York (2002)
Liang, Q.H., Marche, F.: Numerical resolution of well-balanced shallow water equations with complex source terms. Adv. Water Resour. 32, 873–884 (2009)
Rogers, B.D., Borthwick, A.G.L., Taylor, P.H.: Mathematical balancing of flux gradient and source terms prior to using Roe’s approximate Riemann solver. J. Comput. Phys. 168, 422–451 (2003)
Toro, E.F.: Shock-Capturing Methods for Free-Surface Shallow Flows. Wiley, New York (2001)
Zhou, J.G., Causon, D.M., Mingham, C.G., Ingram, D.M.: The surface gradient method for the treatment of source terms in the shallow-water equations. J. Comput. Phys. 168, 1–25 (2001)
Barth, T.J.: Numerical aspects of computing viscous high Reynolds number flows on unstructured meshes. AIAA paper 1991–0721, AIAA 29th ASM, Reno, (1991)
Katz, A., Work, D.: High-order flux correction/finite difference schemes for strand grids. J. Comput. Phys. 282, 360–380 (2015)
Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43, 357–372 (1981)
Diskin, B., Thomas, J.: Comparison of node-centered and cell-centered unstructured finite-volume discretizations: Inviscid fluxes. AIAA paper 2010–1079, AIAA 48th Aerospace Sciences Meeting, Orlando, FL (2010)
Jameson, A., Baker, T., Weatherill, N.: Calculation of inviscid transonic flow over a complete aircraft. AIAA paper 86–0103, AIAA 24th Aerospace Sciences Meeting, Reno, (1986)
Barth, T.J.: A 3-D upwind Euler solver for unstructured meshes. AIAA paper 1991–1548, AIAA 29th ASM, Reno, (1991)
Pincock, B., Katz, A.: High-order flux correction for viscous flows on arbitrary unstructured grids. J. Sci. Comput. 61(2), 454–476 (2014)
Delanaye, M., Liu, Y.: Quadratic reconstruction finite volume schemes on 3D arbitrary unstructured polyhedral grids. AIAA paper 1995–3259, AIAA 14th CFD Conference, Norfolk, (1999)
Delanaye, M.: Polynomial reconstruction finite volume schemes for the compressible euler and navier-stokes equations on unstructured and adaptive grids. Phd thesis, Universite de liege, (1998)
Caughey, D., Jameson, A.: Fast preconditioned multigrid solution of the Euler and Navier-Stokes equations for steady, compressible flows. Int. J. Numer. Meth. Fluids 43, 537–553 (2003)
Shapiro, A.H.: The Dynamics and Thermodynamics of Compressible Fluid Flow, vol. 2. The Ronald Press Company, New York (1954)
Chiocchia, G.: Exact solutions to transonic and supersonic flows. Technical Report AR-211, AGARD, (1985)
Acknowledgments
Development was performed with the support of the Computational Research and Engineering for Acquisition Tools and Environments (CREATE) Program sponsored by the U.S. Department of Defense HPC Modernization Program Office and by the Office of Naval Research Sea-Based Aviation program directed by Dr. Judah Milgram and Mr. John Kinzer.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Thorne, J., Katz, A. Source Term Discretization Effects on the Steady-State Accuracy of Finite Volume Schemes. J Sci Comput 69, 146–169 (2016). https://doi.org/10.1007/s10915-016-0186-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-016-0186-9