Discrete Lie Advection of Differential Forms

R. Abraham, J.E. Marsden, T. Ratiu, Manifolds, Tensor Analysis, and Applications. Applied Mathematical Sciences, vol. 75 (Springer, Berlin, 1988). D.N. Arnold, P.B. Bochev, R.B. Lehoucq, R.A. Nicolaides, M. Shashkov (eds.), Compatible Spatial Discretizations. IMA Volumes, vol. 142 (Springer, Berlin, 2006). D.N. Arnold, R.S. Falk, R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numer. 15, 1–155 (2006). D.N. Arnold, R.S. Falk, R. Winther, Finite element exterior calculus: from Hodge theory to numerical stability, Bull. Am. Math. Soc. 47, 281–354 (2010). P.B. Bochev, J.M. Hyman, Principles of mimetic discretizations of differential operators, IMA 142, 89–119 (2006). A. Bossavit, Computational Electromagnetism (Academic Press, Boston, 1998). A. Bossavit, Extrusion contraction: their discretization via Whitney forms, COMPEL: Int. J. Comput. Math. Electr. Electron. Eng. 22(3), 470–480 (2003). W.L. Burke, Applied Differential Geometry (University Press, Cambridge, 1985). S. Carroll, Spacetime and Geometry: An Introduction to General Relativity (Pearson Education, Upper Saddle River, 2003). É. Cartan, Les Systèmes Differentiels Exterieurs et leurs Applications Géometriques (Hermann, Paris, 1945). M. Desbrun, E. Kanso, Y. Tong, Discrete differential forms for computational sciences, in Discrete Differential Geometry, Course Notes, ed. by E. Grinspun, P. Schröder, M. Desbrun (ACM SIGGRAPH, New York, 2006). T.F. Dupont, Y. Liu, Back-and-forth error compensation and correction methods for removing errors induced by uneven gradients of the level set function, J. Comput. Phys. 190(1), 311–324 (2003). V. Dyadechko, M. Shashkov, Moment-of-Fluid Interface Reconstruction. LANL Technical Report LA-UR-05-7571 (2006). S. Elcott, Y. Tong, E. Kanso, P. Schröder, M. Desbrun, Stable circulation-preserving, simplicial fluids, ACM Trans. Graph. 26(1), 4 (2007). B. Engquist, S. Osher, One-sided difference schemes and transonic flow, PNAS 77(6), 3071–3074 (1980). H. Flanders, Differential Forms and Applications to Physical Sciences (Dover, New York, 1990). T. Frankel, The Geometry of Physics, 2nd edn. (Cambridge University Press, Cambridge, 2004). X. Gu, S.T. Yau, Global conformal surface parameterization, in Symposium on Geometry Processing (2003), pp. 127–137. E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for ODEs (Springer, Berlin, 2002). F.H. Harlow, J.E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surfaces, Phys. Fluids 8, 2182–2189 (1965). H. Heumann, R. Hiptmair, Extrusion contraction upwind schemes for convection-diffusion problems, in Seminar für Angewandte Mathematik SAM 2008-30, ETH Zürich (2008). D.J. Hill, D.I. Pullin, Hybrid tuned center-difference-WENO method for large Eddy simulations in the presence of strong shocks, J. Comput. Phys. 194(2), 435–450 (2004). R. Hiptmair, Finite elements in computational electromagnetism, Acta Numer. 11, 237–339 (2002). A.N. Hirani, Discrete exterior calculus. Ph.D. thesis, Caltech (2003). A. Iske, M. Käser, Conservative semi-Lagrangian advection on adaptive unstructured meshes, Numer. Methods Partial Differ. Equ. 20(3), 388–411 (2004). R.J. LeVeque, CLAWPACK, at http://www.clawpack.org (1994–2009). R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics (Cambridge University Press, Cambridge, 2002). D. Levy, S. Nayak, C. Shu, Y.T. Zhang, Central WENO schemes for Hamilton–Jacobi equations on triangular meshes, J. Sci. Comput. 27, 532–552 (2005). X.D. Liu, S. Osher, T. Chan, Weighted essentially non-oscillatory schemes, J. Sci. Comput. 126, 202–212 (1996). D. Lovelock, H. Rund, Tensors, Differential Forms, and Variational Principles (Dover, New York, 1993). J.E. Marsden, M. West, Discrete mechanics and variational integrators, Acta Numer. (2001). S. Morita, Geometry of Differential Forms. Translations of Mathematical Monographs, vol. 201 (Am. Math. Soc., Providence, 2001). J.R. Munkres, Elements of Algebraic Topology (Addison-Wesley, Menlo Park, 1984). J.C. Nédélec, Mixed Finite Elements in 3D in H(div) and H(curl). Springer Lectures Notes in Mathematics, vol. 1192 (Springer, Berlin, 1986). R.A. Nicolaides, X. Wu, Covolume solutions of three dimensional div–curl equations, SIAM J. Numer. Anal. 34, 2195 (1997). S. Osher, R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces. Applied Mathematical Sciences, vol. 153 (Springer, New York, 2003). J.A. Sethian, Level Set Methods and Fast Marching Methods, 2nd edn. Monographs on Appl. Comput. Math., vol. 3 (Cambridge University Press, Cambridge, 1999). J. Shi, C. Hu, C.W. Shu, A technique for treating negative weights in WENO schemes, J. Comput. Phys. 175, 108–127 (2002). C.W. Shu, Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws. Lecture Notes in Mathematics, vol. 1697 (Springer, Berlin, 1998), pp. 325–432. C.W. Shu, S. Osher, Efficient implementation of essentially non-oscillatory shock capturing schemes, J. Sci. Comput. 77, 439–471 (1988). A. Stern, Y. Tong, M. Desbrun, J.E. Marsden, Variational integrators for Maxwell’s equations with sources, in Progress in Electromagnetics Research Symposium, vol. 4 (2008), pp. 711–715. V.A. Titarev, E.F. Toro, Finite-volume WENO schemes for three-dimensional conservation laws, J. Comput. Phys. 201(1), 238–260 (2004). Y. Tong, P. Alliez, D. Cohen-Steiner, M. Desbrun, Designing quadrangulations with discrete harmonic forms, in Proc. Symp. Geometry Processing (2006), pp. 201–210. H. Whitney, Geometric Integration Theory (Princeton Press, Princeton, 1957). Y.T. Zhang, C.W. Shu, High-order WENO schemes for Hamilton–Jacobi equations on triangular meshes, J. Sci. Comput. 24, 1005–1030 (2003).