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Discrete calculus methods for diffusion. (English) Zbl 1120.65325

Summary: A general methodology for the solution of partial differential equations (PDEs) is described in which the discretization of the calculus is exact and all approximation occurs as an interpolation problem on the material constitutive equations. The fact that the calculus is exact gives these methods the ability to capture the physics of PDE systems well. The construction of both node and cell based methods of first and second-order are described for the problem of unsteady heat conduction – though the method is applicable to any PDE system. The performance of these new methods are compared to classic solution methods on unstructured 2D and 3D meshes for a variety of simple and complex test cases.

MSC:

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
35K05 Heat equation
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[1] Vidovic, D.; Segal, A.; Wesseling, P., A superlinearly convergent finite volume method for the incompressible Navier-Stokes equations, J. Comput. Phys., 198, 159-177 (2004) · Zbl 1057.76044
[2] Wenneker, I.; Segal, A.; Wesseling, P., Conservation properties of a new unstructured staggered scheme, Comput. Fluids, 32, 139-147 (2003) · Zbl 1151.76535
[3] Wenneker, I.; Segal, A.; Wesseling, P., A Mach-uniform unstructured staggered grid method, Int. J. Numer. Meth. Fluids, 40, 1209-1235 (2002) · Zbl 1025.76023
[4] van der Heul, D. R.; Vuik, C.; Wesseling, P., A staggered scheme for hyperbolic conservation laws applied to computation of flow with cavitation, (Toro, E. F., Godunov Methods: Theory and Applications (2001), Kluwer Academic: Kluwer Academic New York), 969-976, 1999 · Zbl 1064.76594
[5] H. Bijl, P. Wesseling, Computation of unsteady flows at all speeds with a staggered scheme, in: E. Oñate, G. Bugeda, B. Suárez (Eds.), ECCOMAS 2000, Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering, Barcelona, September 2000.; H. Bijl, P. Wesseling, Computation of unsteady flows at all speeds with a staggered scheme, in: E. Oñate, G. Bugeda, B. Suárez (Eds.), ECCOMAS 2000, Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering, Barcelona, September 2000.
[6] H. Bijl, Computation of flow at all speeds with a staggered scheme, Ph.D. Thesis, Delft University of Technology, 1999.; H. Bijl, Computation of flow at all speeds with a staggered scheme, Ph.D. Thesis, Delft University of Technology, 1999.
[7] Harlow, F. H.; Welch, J. E., Numerical calculations of time dependent viscous incompressible flow of fluid with a free surface, Phys. Fluids, 8, 12, 2182-2189 (1965) · Zbl 1180.76043
[8] Wesseling, P.; Segal, A.; Kassels, C. G.M.; Bijl, H., Computing flows on general two-dimensional nonsmooth staggered grids, J. Eng. Math., 34, 21-44 (1998) · Zbl 0917.76068
[9] Bijl, H.; Wesseling, P., A unified method for computing incompressible and compressible flows in boundary fitted coordinates, J. Comput. Phys., 141, 153-173 (1998) · Zbl 0918.76054
[10] Nicolaides, R. A., The covolume approach to computing incompressible flow, (Gunzburger, M. D.; Nicolaides, R. A., Incompressible Computational Fluid Dynamics (1993), Cambridge University Press), 234-295 · Zbl 1189.76392
[11] Hyman, J. M.; Shashkov, M., The orthogonal decomposition theorems for mimetic finite difference methods, SIAM J. Num. Anal., 36, 3, 788-818 (1999) · Zbl 0972.65077
[12] Perot, J. B.; Zhang, X., Reformulation of the unstructured staggered mesh method as a classic finite volume method. Reformulation of the unstructured staggered mesh method as a classic finite volume method, Finite Volumes for Complex Applications II (1999), Hermes Science Publications, pp. 263-270 · Zbl 1052.65534
[13] Perot, J. B., Conservation properties of unstructured staggered mesh schemes, J. Comput. Phys., 159, 58-89 (2000) · Zbl 0972.76068
[14] Perot, J. B.; Nallapati, R., A moving unstructured staggered mesh method for the simulation of incompressible free-surface flows, J. Comput. Phys., 184, 192-214 (2003) · Zbl 1118.76307
[15] Lilly, D. K., On the computational stability of numerical solutions of time-dependent non-linear geophysical fluid dynamics problems, Mon. Weather Rev., 93, 1, 11-26 (1965)
[16] Zhang, X.; Schmidt, D.; Perot, J. B., Accuracy and conservation properties of a three-dimensional unstructured staggered mesh scheme for fluid dynamics, J. Comput. Phys., 175, 764-791 (2002) · Zbl 1018.76036
[17] Chang, W.; Giraldo, F.; Perot, J. B., Analysis of an exact fractional step method, J. Comput. Phys., 179, 1-17 (2002)
[18] Mattiussi, C., An analysis of finite volume, finite element, and finite difference methods using some concepts from algebraic topology, J. Comput. Phys., 133, 289-309 (1997) · Zbl 0878.65091
[19] Subramanian, V.; Perot, J. B., Higher order mimetic methods for unstructured meshes, J. Comput. Phys., 219, 68-85 (2006) · Zbl 1105.65101
[20] Nicolaides, R.; Wu, X., Covolume solutions of three-dimensional div-curl equations, SIAM J. Num. Anal., 34, 2195-2203 (1997) · Zbl 0889.35006
[21] R.A. Nicolaides, Direct discretization of planar div-curl problems, ICASE Report 89-76, 1989.; R.A. Nicolaides, Direct discretization of planar div-curl problems, ICASE Report 89-76, 1989.
[22] Perot, J. B.; Vidovic, D.; Wesseling, P., Mimetic reconstruction of vectors, (Arnold, D. N.; Bochev, P. B.; Lehoucq, R. B.; Nicolaides, R. A.; Shashkov, M., IMA Volumes in Mathematics and its Applications. IMA Volumes in Mathematics and its Applications, Compatible Spatial Discretizations, vol. 142 (2006), Springer: Springer New York) · Zbl 1110.65108
[23] Chang, W.; Giraldo, F.; Perot, J. B., Analysis of an exact fractional step method, J. Comput. Phys., 179, 1-17 (2002)
[24] M. Desbrun, A. Hirani, M. Leok, J. Marsden, Discrete Exterior Calculus, www.citebase.org/abstract?id=oai:arXiv.org:math/0508341; M. Desbrun, A. Hirani, M. Leok, J. Marsden, Discrete Exterior Calculus, www.citebase.org/abstract?id=oai:arXiv.org:math/0508341 · Zbl 1080.39021
[25] de Rham, G., Sut l’analysis situs des varietes a n dimensions, J. Math., 10, 115-199 (1931) · Zbl 0002.05502
[26] Bossavit, A.; Mayergoyz, I., Edge elements for scattering problems, IEEE Trans. Mag., 25, 4, 2816-2821 (1989)
[27] D. White, Orthogonal vector basis functions for time domain finite element solution of the vector wave equation, in: Proceedings of the 8th Biennial IEEE Conference on Electromagnetic Field Computation, Tucson, AZ, 1998, UCRL-JC-129188.; D. White, Orthogonal vector basis functions for time domain finite element solution of the vector wave equation, in: Proceedings of the 8th Biennial IEEE Conference on Electromagnetic Field Computation, Tucson, AZ, 1998, UCRL-JC-129188.
[28] P. Castillo, J. Koning, R.Rieben, M. Stowell, D. White, Discrete Differential Forms: A Novel Methodology for Robust Computational Electromagnetics, LLNL Report UCRL-ID-151522, January 2003.; P. Castillo, J. Koning, R.Rieben, M. Stowell, D. White, Discrete Differential Forms: A Novel Methodology for Robust Computational Electromagnetics, LLNL Report UCRL-ID-151522, January 2003.
[29] Rodrigue, G.; White, D., A vector finite element time domain method for solving Maxwells equations on unstructured hexahedral grids, SIAM J. Sci. Comput., 23, 3, 683-706 (2001) · Zbl 1003.78008
[30] T.J. Barth, P.O. Frederickson, Higher Order Solution of the Euler Equations on Unstructured Grids Using Quadratic Reconstruction, AIAA 90-0013, 1990.; T.J. Barth, P.O. Frederickson, Higher Order Solution of the Euler Equations on Unstructured Grids Using Quadratic Reconstruction, AIAA 90-0013, 1990.
[31] T.J. Barth, Recent Improvements in High Order K-exact Reconstruction on Unstructured Meshes, AIAA 93-0668, 1993.; T.J. Barth, Recent Improvements in High Order K-exact Reconstruction on Unstructured Meshes, AIAA 93-0668, 1993.
[32] Sukumar, N., Voronoi cell finite difference method for the diffusion operator on arbitrary unstructured grids, Int. J. Num. Meth. Eng., 57, 1-34 (2003) · Zbl 1042.65079
[33] Cavendish, J. C.; Hall, C. A.; Porsching, T. A., A complementary volume approach for modeling three-dimensional Navier-Stokes equations using dual Delaunay/Voronoi tessellations, Int. J. Num. Meth. Heat Fluid Flow, 4, 329-345 (1994) · Zbl 0815.76041
[34] Hall, C. A.; Peterson, J. S.; Porsching, T. A.; Sledge, F. R., The dual variable method for finite element discretizations of Navier/Stokes equations, Int. J. Num. Meth., 21, 883-898 (1985) · Zbl 0587.76045
[35] Shashkov, M.; Steinberg, S., Solving diffusion equations with rough coefficients in rough grids, J. Comput. Phys., 129, 383-405 (1996) · Zbl 0874.65062
[36] Morel, J. M.; Dendy, J. E.; Hall, M. L.; White, S. W., A cell-centered Lagrangian-mesh diffusion differencing scheme, J. Comput. Phys., 103, 286 (1992) · Zbl 0763.76052
[37] Wesseling, P., Principals of Computational Fluid Dynamics (2001), Springer: Springer Berlin, ISBN 3-540-67853-0
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