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A review of numerical methods for nonlinear partial differential equations


Author: Eitan Tadmor
Journal: Bull. Amer. Math. Soc. 49 (2012), 507-554
MSC (2010): Primary 35J60, 35K55, 35L65, 35L70, 65Mxx, 65Nxx
DOI: https://doi.org/10.1090/S0273-0979-2012-01379-4
Published electronically: July 20, 2012
MathSciNet review: 2958929
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Abstract: Numerical methods were first put into use as an effective tool for solving partial differential equations (PDEs) by John von Neumann in the mid-1940s. In a 1949 letter von Neumann wrote ``the entire computing machine is merely one component of a greater whole, namely, of the unity formed by the computing machine, the mathematical problems that go with it, and the type of planning which is called by both.'' The ``greater whole'' is viewed today as scientific computation: over the past sixty years, scientific computation has emerged as the most versatile tool to complement theory and experiments, and numerical methods for solving PDEs are at the heart of many of today's advanced scientific computations. Numerical solutions found their way from financial models on Wall Street to traffic models on Main Street. Here we provide a bird's eye view on the development of these numerical methods with a particular emphasis on nonlinear PDEs.


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  • 1. Adams, R. and Fournier, J. (2003), Sobolev spaces, 2nd ed., Academic Press. MR 2424078 (2009e:46025)
  • 2. Alvarez, L., Lions, P. L. and Morel J. M. (1992), Image selective smoothing and edge detection by non-linear diffusion II, SIAM J. Num. Anal., Vol. 29 (3), pp. 845-866. MR 1163360 (93a:35072)
  • 3. Arakawa, A. (1966), Computational design for long-term integration of the equations of fluid motion: two-dimensional incompressible flows. I. J.Comput. Phys. v. 1, pp. 119-143 (reprinted in v. 135, pp. 103-114) MR 1486265
  • 4. Arnold, D. N. (1990), Mixed finite element methods for elliptic problems, Comput. Methods Appl. Mech. Engrg., 82(1-3), pp. 281-300. MR 1077658 (91h:65168)
  • 5. Arnold, D. N., Falk, R. S. and Winther, R. (2006), Finite element exterior calculus, homological techniques, and applications, Acta Numer. 15, pp. 1-155. MR 2269741 (2007j:58002)
  • 6. Arnold, D. N., Brezzi, F., Cockburn B., and Marini, L. D. (2002), Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39(5), pp. 1749-1779. MR 1885715 (2002k:65183)
  • 7. Artstein, Z., Gear, C. W., Kevrekidis, I. G., Slemrod M., and Titi, E. S. (2011), Analysis and computation of a discrete KdV-Burgers type equation with fast dispersion and slow diffusion, SIAM J. Numer. Anal., v. 49(5), pp. 2124-2143. MR 2861712
  • 8. Babuška, I. (1971), Error Bounds for Finite Element Method, Numerische Mathematik, v. 16, pp. 322-333. MR 0288971 (44:6166)
  • 9. Babuška, I. (1973), The finite element method with Lagrangian multipliers, Numer. Math. v20, pp. 179-192. MR 0359352 (50:11806)
  • 10. Babuška, I. and Suri, M. (1994), The $ p$ and $ hp$ versions of the finite element method, basis principles and properties, SIAM Review, v. 36, pp. 578-632. MR 1306924 (96d:65184)
  • 11. Balbas, J. and Tadmor, E. (2010), CentPack: A package of high-resolution central schemes for non-linear conservation laws and related problems, http://www.cscamm.umd.edu/centpack/.
  • 12. Bao, W., Du, Q. and Zhang, Y. (2006), Dynamics of rotating Bose-Einstein condensates and its efficient and accurate numerical computation, SIAM J. Appl. Math. 66 758-786. MR 2216159 (2006k:35267)
  • 13. Bao, W., Li, H.-L, and Shen, J. (2009), A generalized-Laguerre-Fourier-Hermite pseudospectral method for computing the dynamics of rotating Bose-Einstein condensates, SIAM J. Sci. Comput. 31, pp. 3685-3711. MR 2556558 (2010j:65193)
  • 14. Becker, R. and Rolf Rannacher, R. (2001), An optimal control approach to a posteriori error estimation in finite element methods, Acta Numerica v. 10, pp 1-102. MR 2009692 (2004g:65147)
  • 15. Berger, M. and Oliger, J. (1984), Adaptive mesh refinement for hyperbolic partial differential equations, J. Comput. Phys., v. 53, pp. 484-512. MR 739112 (85h:65211)
  • 16. Bensoussan, A., Lions, J.-L. and Papanicolaou, G. (1978), Asymptotic analysis for periodic structures, Studies in Mathematics and its applications, Vol. 5, North-Holland Publishing Company. MR 503330 (82h:35001)
  • 17. Brebbia, C. A., (1978), The Boundary Element Method for Engineers, Pentech Press/Halstead. MR 806953 (87b:65192)
  • 18. Brebbia, C. A. and Dominguez, J. (1977), Boundary element methods for potential problems, J. Appl. Math. Model. v. 1, pp 372-378. MR 0520348 (58:25017)
  • 19. Bernardi, C. and Maday, M. (1992), Approximations spectrales de problemes aux limites elliptiques Springer-Verlag, Paris. MR 1208043 (94f:65112)
  • 20. Bernardi, C. and Maday, M. (1996), Spectral Element Methods, (P. G. Ciarlet and J. L. Lions, eds), Handbook of Numerical Analysis, North-Holland, Amsterdam.
  • 21. Bianchini, S. and Bressan, A. (2005), Vanishing viscosity solutions of non-linear hyperbolic systems. Ann. of Math. (2) 161, no. 1, pp. 223-342. MR 2150387 (2007i:35160)
  • 22. Binev, P., Dahmen, W. and DeVore R. (2004), Adaptive finite element methods with convergence rates, Numer. Math., Vol 97(2), pp. 219-268. MR 2050077 (2005d:65222)
  • 23. Bochev, P. and Gunzburger, M. (2006), Least-squares finite element methods, ICM 2006. Invited Lectures. Abstracts. Section 16 (Sanz-Solé M., Soria J., Varona J. L. & Verdera J. eds.), vol. III. pp. 1137-1162, EMS. MR 2275722 (2008d:65131)
  • 24. Brandt, A. (1977), Multi-level adaptive solutions to boundary-value problems, Math. Comp, 31, pp. 333-390. MR 0431719 (55:4714)
  • 25. Brenner, S. and Scott, R. L. (2005), The Mathematical Theory of Finite Element Methods, 2nd edition, Springer.
  • 26. Brezzi, F. (1974), On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge v. 8(2), pp. 129-151. MR 0365287 (51:1540)
  • 27. Brezzi, F. and Fortin, M. (1991), Mixed and Hybrid Finite Element Methods, Springer Verlag. MR 1115205 (92d:65187)
  • 28. Butcher, J. C. (2003), Numerical Methods for Ordinary Differential Equations, Wiley, 2003. MR 1993957 (2004e:65069)
  • 29. Caffarelli, L. (2002), Non-linear elliptic theory and the Monge-Ampere equation, ``International Congress of Mathematicians'', Proceedings of the ICM02 Beijing 2002 (Li Tatsien, ed.), Vol. I: Plenary lectures, Higher Education Press (2002) 179-187. MR 1989184 (2004d:35068)
  • 30. Caffarelli, L. and Cabré, X. (1995), Fully Non-linear Elliptic Equations, Colloq. Publ. Vol. 43, Amer. Math. Soc. MR 1351007 (96h:35046)
  • 31. Caffarelli, L. and Souganidis, P. (2007), A rate of convergence for monotone finite difference approximations to fully non-linear, uniformly elliptic PDEs, Comm. Pure Appl. Math., v. 61(1), pp. 1-17. MR 2361302 (2008m:65288)
  • 32. Candés, E. J. and Donoho, D. L. (2001), Curvelets and curvilinear integrals. J. Approx. Theory 113(1), pp. 59-90. MR 1866248 (2002j:41012)
  • 33. Candés, E. J., Romberg, J. and Tao, T. (2006), Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inform. Theory 52(2), pp. 489-509. MR 2236170 (2007e:94020)
  • 34. do Carmo., M. P. (1976), Differential Geometry of Curves and Surfaces, Prentice Hall. MR 0394451 (52:15253)
  • 35. Cercignani, C. (1998), The Boltzmann Equation and Its Applications, Springer Verlag. MR 1313028 (95i:82082)
  • 36. Chan T. F., Xu, J. and Zikatanov, L. (1998), An agglomeration multigrid method for unstructured grids, Contemp. Math. v. 218, pp. 67-81. MR 1645844 (99h:65205)
  • 37. Chang, S.-Y. A. and Yang, P. C. (2002), Non-linear partial differential equations in conformal geometry, Proc. Inter. Congress Math. Vol. I (Beijing, 2002), pp. 189-207, Higher Ed. Press, Beijing. MR 1989185 (2004d:53031)
  • 38. Charney, J. G., Fjörtoft R. and von Neumann, J. (1950), Numerical Integration of the Barotropic Vorticity Equation, Tellus, Vol. 2, pp. 237-254. MR 0042799 (13:164f)
  • 39. Chertock, A. and Levy, D. (2002), Particle methods for the KdV equation, J. Sci. Comput. Vol. 17, pp. 491-499. MR 1910746
  • 40. Chorin, A. J. (1968), Numerical Solution of the Navier-Stokes Equations, Math. Comp., 22, pp. 745. MR 0242392 (39:3723)
  • 41. Chorin, A. J (1973), Numerical study of slightly viscous flow, Journal of Fluid Mechanics, v. 57, pp. 785-796. MR 0395483 (52:16280)
  • 42. Chorin, A. and Hald, O. (2006), Stochastic Tools in Mathematics and Sciences, Surveys Tutorials Appl. Math. Sci, vol. 1, Springer. MR 2189824 (2006j:60001)
  • 43. Ciarlet, P. G. (1978), The Finite Element Method for Elliptic Problems, North-Holland. MR 0520174 (58:25001)
  • 44. Ciarlet, P. G. and Lions, J. L., eds. (1991), Handbook of Numerical Analysis, Vol. II: Finite Element Methods (1991) and Vol. IV: Finite Element Methods, Numerical Methods for Solids, (1996), North-Holland, Amsterdam. MR 1115235 (92f:65001)
  • 45. Ciarlet, P. G. and Lions, J. L., eds. (2000), Handbook of Numerical Analysis, Vol. VII: Finite Volume Methods, North-Holland, Amsterdam. MR 1804744 (2001h:65001)
  • 46. Clay Mathematics Institute, (2000), Millennium Problems, http://www.claymath. org/millennium/.
  • 47. Cockburn, B. (1998), An introduction to the discontinuous Galerkin method for convection-dominated problems, Advanced numerical approximation of nonlinear hyperbolic equations (Cetraro, 1997), Lecture Notes in Math., Springer, Berlin., Vol. 1697, pp. 151-268. MR 1728854 (2001f:65117)
  • 48. Cockburn, B., Coquel, F. and LeFloch, P. G. (1995), Convergence of finite volume methods for multidimensional conservation laws, SIAM J. Numer. Anal. 32, pp. 687-705. MR 1335651 (97f:65051)
  • 49. Cockburn, B., Johnson, C., Shu, C.-W. and Tadmor, E. (1998), Approximate solutions of nonlinear conservation laws, in "Advanced Numerical Approximation of Nonlinear Hyperbolic Equations" C.I.M.E. course in Cetraro, Italy, June 1997 (A. Quarteroni ed.), Lecture notes in Mathematics v. 1697, Springer Verlag. MR 1729305 (2000h:65004)
  • 50. Colella, P. and Woodward, P. (1984), The piecewise parabolic method (PPM) for gas-dynamical simulations, J. Comput. Physics 54, 174-201.
  • 51. Constantin, P. and Foias, C. (1988), Navier-Stokes Equations, The Univ. of Chicago Press. MR 972259 (90b:35190)
  • 52. Cooley, J. W. and Tukey, J. W. (1965), An algorithm for the machine calculation of complex Fourier series, Math. Comput. v. 19, pp. 297-301. MR 0178586 (31:2843)
  • 53. Coquel, F. and LeFloch, P. G. (1993), Convergence of finite difference schemes for scalar conservation laws in several space variables. General theory, SIAM J. Numer. Anal. 30 pp. 675-700. MR 1220646 (94e:65092)
  • 54. Courant, R. (1943), Variational methods for the solution of problems of equilibrium and vibrations, Bull. AMS 49 pp. 1-23. MR 0007838 (4:200e)
  • 55. Courant, R., Friedrichs, K. and Lewy, H. (1967), On the partial difference equations of mathematical physics, IBM Journal, vol. 11(2) pp. 215-234, English translation of the 1928 German original ``Über die partiellen Differenzengleichungen der mathematischen Physik'', Mathematische Annalen, Vol. 100, No. 1, pp. 32-74, 1928. MR 1512478
  • 56. Courant, R. and Friedrichs, K.-O. (1948), Supersonic Flow and Shock Waves, Springer, New York. MR 0029615 (10:637c)
  • 57. Crandall, M. G. and Lions, P.-L. (1983), Viscosity Solutions of Hamilton-Jacobi Equations, Transactions of the Amer. Math. Soc., v. 277(1), pp. 1-42. MR 690039 (85g:35029)
  • 58. Crouzeix, M. and Raviart, P. A. (1973), Conforming and non-conforming finite element methods for solving the stationary Stokes equations, R.A.I.R.O. Anal. Numer. v. 7 pp. 33-76. MR 0343661 (49:8401)
  • 59. Dafermos, C. (2010), Hyperbolic Conservation Laws in Continuum Physics, v. 325, 3rd edition, Springer. MR 2574377 (2011i:35150)
  • 60. Dal Maso, G. (1993), An Introduction to G-convergence. Birkhäuser, Basel.
  • 61. Daubechies, I. (1992), Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics. MR 1162107 (93e:42045)
  • 62. Dautray, R. and Lions, J.-L. (2000), Mathematical Analysis and Numerical Methods for Science and Technology, vol. 4., Springer. MR 1081946 (91h:00004b)
  • 63. Dawson, J. M. (1983), Particle simulation of plasmas, Reviews of Modern Physics v. 55, pp. 403.
  • 64. Dean, E. J. and Glowinski, R. (2004), Numerical solution of the two-dimensional elliptic Monge-Ampére equation with Dirichlet boundary conditions: a least-squares approach, Comptes Rendus Mathematique Vol. 339(12), pp. 887-892. MR 2111728
  • 65. Degond, P. and Mas-Gallic, S. (1991), The weighted particle method for convection-difussion equations, Parts 1 & 2, Math. Comput. Vol. 53, pp. 485. MR 983559 (90g:65126)
  • 66. DeVore, R. A. (2007), Optimal computation. International Congress of Mathematicians. Vol. I, pp. 187-215, Eur. Math. Soc., Zürich. MR 2334191 (2008h:41005)
  • 67. DiPerna, R. J. and Lions, P. L. (1989), On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. Math. 130, 1989, 321-366. MR 1014927 (90k:82045)
  • 68. DiPerna, R. J., Lions, P.-L. and Meyer, Y. (1991), $ L^p$ regularity of velocity averages Ann. Inst. H. Poincaré Anal. Non Lin. 8, pp. 271-288. MR 1127927 (92g:35036)
  • 69. Donoho, D. L. (2006), Compressed sensing. IEEE Trans. Inform. Theory 52 (2006), no. 4, pp. 1289-1306. MR 2241189 (2007e:94013)
  • 70. Douglas, J. (1931), Solution of the problem of Plateau, Trans. Amer. Math. Soc. v. 33, pp. 263-321. MR 1501590
  • 71. E, W. and Engquist, B. (2003), The heterogeneous multiscale methods, Commun. Math. Sci. 1, pp. 87-132. MR 1979846 (2004b:35019)
  • 72. Efendiev, Y. and Hou, T.-Y. (2009), Multiscale Finite Element Methods. Theory and Applications, Surveys and Tutorials in the Appl. Math. Sci., v4, Springer. MR 2477579 (2010h:65224)
  • 73. Ekeland, I. and Témam, R. (1999), Convex Analysis and Variational Problems, Classics in Appl. Math., v28, SIAM. MR 1727362 (2000j:49001)
  • 74. Engquist, B. and Runborg, O. (2003), Computational high frequency wave propagation, Acta Numerica, v. 12, pp. 181-266. MR 2249156 (2007f:65043)
  • 75. Evans, L. (1988), Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS, Amer. Math. Soc. 74. MR 1034481 (91a:35009)
  • 76. Evans, L. (1998), Partial Differential Equations, Grad. Text Math Vol. 19. Amer. Math. Soc. MR 1625845 (99e:35001)
  • 77. Eymard, R., Gallouët, T. and Herbin, R. (2001), Finite volume approximation of elliptic problems and convergence of an approximate gradient, Applied Numerical Math. v. 37(1-2), pp. 31-53. MR 1825115 (2003c:65118)
  • 78. Filbet, F. (2006), A finite volume scheme for the Patlak-Keller-Segel chemotaxis model, Numer. Math. 104(4), pp. pp. 457-488. MR 2249674 (2007e:92002)
  • 79. Fjordholm, U., Mishra, S. and Tadmor, E. (2012), ENO reconstruction and ENO interpolation are stable, Found, of Comp. Math., in press.
  • 80. Francis, J. G. F. (1961), The QR transformation: a unitary analogue to the LR transformation. I., and The QR transformation. II., Comput. v. 4, pp. 265-271 and Comput. v. 4, pp. 332-345. MR 0130111 (23:B3143)
  • 81. Friedman, A. (1964), Partial Differential Equations of Parabolic Type, Prentice Hall. MR 0181836 (31:6062)
  • 82. Gear, C. W. (1973), Numerical Initial Value Problems in Ordinary Differential Equations. Prentice-Hall, Englewood Cliffs, NJ. MR 0315898 (47:4447)
  • 83. Gelfand, I. M. and Fomin, S. V. (1963), Calculus of Variations, Dover. MR 0160139 (28:3353)
  • 84. Gérard, P. (1991), Microlocal defect measures, Comm. Partial Differential Eqs 16, pp. 1761-1794. MR 1135919 (92k:35027)
  • 85. Giles, M. B. and Süli, E. (2002), Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality, Acta Numerica, Vol. 11, pp. 145-236. MR 2009374 (2005d:65190)
  • 86. Glimm, J. (1965), Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure App. Math., 18, pp. 697-715. MR 0194770 (33:2976)
  • 87. Glimm, J., Grove, J. W., Li, X. L. and Zhao, N. (1999), Simple front tracking, Contmp. Math., v.238, pp. 133-149. MR 1724660 (2000h:76135)
  • 88. Gilbarg, D. and Trudinger, N. S. (1977), Elliptic Partial Differential Equations of Second Order, Springer-Verlag. MR 0473443 (57:13109)
  • 89. Godunov, S. K. (1959), A difference scheme for numerical computation of discontinuous solutions of equations of fluids dynamics, Math. Sb., Vol. 47, pp. 271-290. MR 0119433 (22:10194)
  • 90. Godlewski, E. and Raviart, P.-A. (1996), Numerical Approximation of Hyperbolic Systems of Conservation Laws, Appl. Math Sci. v 118, Springer. MR 1410987 (98d:65109)
  • 91. Golse, F., Lions, P-L., Perthame, B. and Sentis, R. (1988), Regularity of the moments of the solution of a transport equation, J. Funct. Anal. 76, pp. 110-125. MR 923047 (89a:35179)
  • 92. Golub, G. H. and Van Loan, C. (1996), Matrix Computations, 3rd Edition, Johns Hopkins Studies in Mathematical Sciences. MR 1417720 (97g:65006)
  • 93. Gottlieb, D. and Orszag, S. (1977), Numerical analysis of spectral methods: theory and applications, CBMS-NSF Regional Conference Ser. Appl. Math., SIAM, Philadelphia, 1977. MR 0520152 (58:24983)
  • 94. Greengard, L. and Rokhlin, V. (1987), A fast algorithm for particle simulations, Journal of Computational Physics, 73(2), pp. 325-348. MR 918448 (88k:82007)
  • 95. Gustasfsson, B., Kreiss, H.-O., and Oliger, J. (1995), Time dependent problems and difference methods, Wiley-interscinece. MR 1377057 (97c:65145)
  • 96. Hairer, E., Nørsett, S. P. and Wanner, G. (1993), Solving ordinary differential equations I: Nonstiff problems, second edition, Springer Verlag, Berlin. MR 1227985 (94c:65005)
  • 97. Hairer, E. and Wanner, G. (1996), Solving ordinary differential equations II: Stiff and differential-algebraic problems, Springer Verlag, Berlin. MR 1439506 (97m:65007)
  • 98. Hamilton, R. S. (1982), Three Manifolds with Positive Ricci Curvature, J. Diff. Geom. Vol. 17, pp. 255-306. MR 664497 (84a:53050)
  • 99. Han, Q. and Lin, F.-H. (1997), Elliptic Partial Differential Equations, Courant Lecture Notes, v.1, Amer. Math. Soc., Providence. MR 1669352 (2001d:35035)
  • 100. Harten, A. (1983), High resolution schemes for hyperbolic conservation laws. J. Comput. Phys., v. 49, pp. 357-393. MR 701178 (84g:65115)
  • 101. Harten, A. (1989), ENO schemes with subcell resolution, J. Comput. Phys. v. 83(1), pp. 148-184. MR 1010163 (90i:76010)
  • 102. Harten, A., Engquist, B., Osher, S. and Chakravarthy, S. R. (1987), Uniformly high order accurate essentially non-oscillatory schemes. III, J. Comput. Physics 71, pp. 231-303. MR 897244 (90a:65199)
  • 103. Hesthaven, J. S., Gottlieb, S. and Gottlieb, D. (2007), Spectral Methods for Time-Dependent Problems, Cambridge Monographs on Applied Comput. Math. v. 21, Cambridge Univ. Press, Cambrigde, UK. MR 2333926 (2008i:65223)
  • 104. Hillen, T. and Painter, K. J. (2009), A User's Guide to PDE Models for Chemotaxis, J. Math. Biology, v. 58, pp. 183-217. MR 2448428 (2009m:92017)
  • 105. Holden, H., and Risbero, N. H. (2002), Front Tracking for Hyperbolic Conservation Laws, Springer. MR 1912206 (2003e:35001)
  • 106. Horstman, D. (2003), From 1970 until now: The Keller Segel model in chemotaxis and its consequences I, Jahresber DMV 105, pp. 103-169 and (2004) 106, pp. 51-69. MR 2073515 (2005b:92005)
  • 107. Hou, T. Y. and Lax, P. (1991), Dispersive approximations in fluid dynamics, Comm. Pure and Appl. Math., v. 44, pp. 1-40. MR 1077912 (91m:76088)
  • 108. Hou, T. Y., Li, C., Shi, Z., Wang, S. and Yu, X. (2011), On dingularity formation of a nonlinear nonlocal system, Arch. Rat. Mech. Anal. v. 199, pp. 117-144. MR 2754339 (2012c:35184)
  • 109. Hou, T. Y. and Wu, X. H. (1997), A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys., v. 134, pp. 169-189. MR 1455261 (98e:73132)
  • 110. Hughes, T. J. R. (2000), The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Dover Civil and Mechanical Engineering. MR 1008473 (90i:65001)
  • 111. Hwu W. W.-M. (2011), GPU Computing Gems, Elseveir.
  • 112. Itô, K. (1983), Foundations of stochastic differential equations in infinite dimensional spaces, CBMS Notes, SIAM, Baton Rouge. MR 771478 (87a:60068)
  • 113. Jaswon, M. A. (1963), Integral equation methods in potential theory. I, Proc. Royal Soc. London Series A, v. 275(1360) pp. 23-32; MR 0154075 (27:4034)
  • 114. Johnson, C. (1988), Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, New York. MR 925005 (89b:65003a)
  • 115. Johnson, C. and Thomée, V. (1975), Error Estimates for a Finite Element Approximation of a Minimal Surface, Math. Comp. v.29, pp. 343-349. MR 0400741 (53:4571)
  • 116. Kalnay, E. (2002), Atmospheric Modeling, Data Assimilation and Predictability, Cambridge University Press.
  • 117. Karmarkar, N. (1984), A New Polynomial Time Algorithm for Linear Programming, Combinatorica, Vol 4, nr. 4, p. 373-395. MR 779900 (86i:90072)
  • 118. Karatzas, I. and Shreve, S. E. (1998), Methods of Mathematical Finance. Applications of Mathematics, v. 39. Springer-Verlag, New York. MR 1640352 (2000e:91076)
  • 119. Keller, E. F. and Segel, L. A. (1971), Model for chemotaxis. J. Theor Biol, 30(2), pp. 225-234.
  • 120. Kevrekidis, I. G., Gear, W., Hyman, J. M., Kevrekidis, G., Runborg O. and Theodoropoulos, C. (2003), Equation-Free Multiscale Computation: enabling microscopic simulators to perform system-level tasks, Comm. Math. Sciences 1(4) pp. 715-762. MR 2041455 (2005a:65075)
  • 121. Kohn, R. V. (2007), Energy-driven pattern formation. Inter. Congress Math. Vol. I, pp. 359-383, Eur. Math. Soc., Zürich. MR 2334197 (2008k:49049)
  • 122. Kreiss, H.-O. (1964), On difference approximations of the dissipative type for hyperbolic differential equations, Comm. Pure Appl. Math. v. 17, pp. 335-353. MR 0166937 (29:4210)
  • 123. Kreiss H.-O. and Oliger, J. (1972), Comparison of Accurate Methods for the Integration of Hyperbolic Equations, Tellus, v. 24, pp. 199-215. MR 0319382 (47:7926)
  • 124. Kreiss, H.-O. and Lorenz, J. (2004), Initial-Boundary Value Problems and the Navier-Stokes Equations, Academic Press. MR 998379 (91a:35138)
  • 125. Krylov, N. V. (1997), Lectures on Elliptic and Parabolic Equations in Holder Spaces, Graduate Studies in Math., v. 12, Amer. Math. Soc., Providence. MR 1406091 (97i:35001)
  • 126. Kurganov, A. and Tadmor, E. (2000), New high resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Physics, v. 160, pp. 241-282. MR 1756766 (2001d:65135)
  • 127. Küther, M. and Ohlberger, M. (2003), Adaptive Second Order Central Schemes on Unstructured Staggered Grids, Proc. Ninth Intl. Conference Hyperbolic Problems, (T. Hou and E. Tadmor, eds), Springer-Verlag, pp. 675-684. MR 2053216
  • 128. Lax, P. D. (1954), Weak solution of nonlinear hyperbolic equations and their numerical computation, Comm. Pure Appl. Math. 7, pp. 159-193. MR 0066040 (16:524g)
  • 129. Lax, P. D. (1978), Accuracy and resolution in the computation of solutions of linear and nonlinear equations, ``Recent advances in Numerical Analysis'', Proc. Symp. Math. Res. Ctr., Univ. of Wisconsin, Academic Press, pp. 107-117. MR 519059 (80b:65147)
  • 130. Lax, P. D. (1986), On dispersive difference schemes, Phys. D, 18, pp. 250-254. MR 838330 (87h:65155)
  • 131. Lax, P. D. (2006), Hyperbolic Partial Differential Equations, Courant Lecture Notes, v.14, Amer. Math. Soc., Providence. MR 2273657 (2007h:35002)
  • 132. Lax, P. D. and Nirenberg, L. (1966), On stability for difference schemes: A sharp form of Garding's inequality, Comm. Pure Appl. Math. v. 19, pp. 473-492. MR 0206534 (34:6352)
  • 133. Lax, P. D. and Richtmyer, R. D. (1956), Survey of the stability of linear finite difference equations. Comm. Pure Appl. Math. 9, pp. 267-293. MR 0079204 (18:48c)
  • 134. Lax, P. D. and Wendroff, B. (1960), Systems of conservation laws, Commun. Pure Appl Math. v. 13(2), pp. 217-237. MR 0120774 (22:11523)
  • 135. van Leer, B. (1979), Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method, J. Comput. Phys. 32, 101-136.
  • 136. LeVeque, R. J. (2002), Finite Volume Methods for Hyperbolic Problems, Cambridge texts in Appl. Math. MR 1925043 (2003h:65001)
  • 137. Levy, D. and Tadmor, E (1997), Non-oscillatory central schemes for the incompressible 2-D Euler equations, Mathematical Research Letters 4 (3), 1997, 321-340. MR 1453063 (99a:65100)
  • 138. Lions, P. L. (1984), The concentration-compactness principle in the calculus of variations. The locally compact case, part 1. Annales de l'institut Henri Poincaré (C) Analyse non linéaire, 1 no. 2, pp. 109-145. MR 778970 (87e:49035a)
  • 139. Lions, P. L. (1996), Mathematical Topics in Fluid Mechanics: Volume 1 - Incompressible Models, Oxford Lecture Series in Mathematics and Its Applications. MR 1422251 (98b:76001)
  • 140. Lions, P. L. (1998), Mathematical Topics in Fluid Mechanics: Volume 2 -Compressible Models, Oxford Lecture Series in Mathematics and Its Applications, Oxford University Press. MR 1637634 (99m:76001)
  • 141. Liu, X.-D., Osher, S. and Chan, T. (1994), Weighted essentially non-oscillatory schemes. J. Comput. Phys. v. 115(1), pp. 200-212. MR 1300340
  • 142. Liu, Y.-J., Shu, C.-W., Tadmor E. and Zhang, M. (2007), Central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction, SIAM J. Numer. Anal., Vol. 45(6), pp. 2442-2467 MR 2361897 (2009a:65256)
  • 143. Majda, A., McDonough, J. and Osher, S. (1978), The Fourier method for nonsmooth initial data, Math. Comp. 32(144) pp. 1041-1081. MR 501995 (80a:65197)
  • 144. Majda, A. and Bertozzi A. (2001), Vorticity and Incompressible Flow, Cambridhe texts Appl. Math.
  • 145. Majda, A. (2003), Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes, vol 9., AMS. MR 1965452 (2004b:76152)
  • 146. Majda, A., Abramov, R. V. and Grote, M. J. (2005), Information Theory and Stochastics for Multiscale Nonlinear Systems, CRM monograph series 25, American Mathematical Society. MR 2166171 (2006k:76110)
  • 147. Mas-Gallic, S. and Raviart, P.-A. (1987), A particle method for first order symmetric systems, Numer. Math. 51, pp. 323-352. MR 895090 (88d:65132)
  • 148. MATLAB - The Language of Technical Computing, http://www.mathworks.com/products/ matlab/.
  • 149. Meyer, Y. (1993), Wavelets: Algorithms and Applications, Society Industrial Appl. Math. (SIAM), Philadelphia, PA. MR 1219953 (95f:94005)
  • 150. Meyers, N. and Serrin, J. (1964), $ H=W$, Proc. Nat, Acad. Sci. USA 51, pp. 1055-1056. MR 0164252 (29:1551)
  • 151. Mock, M. S. and Lax, P. D. (1978), The computation of discontinuous solutions of linear hyperbolic equations, Comm. Pure Appl. Math. Vol. 31, pp. 423-430. MR 0468216 (57:8054)
  • 152. Moore, G. E. (1965), Cramming more components onto integrated circuits, Electronics, v. 38 (8).
  • 153. Morgan, J. and Tian G. (2007), Ricci Flow and the Poincare Conjecture, Clay Mathematics Monographs. MR 2334563 (2008d:57020)
  • 154. Murry, J. D. (2002), Mathematical Biology, Vols I & II, Springer.
  • 155. Nessyahu, H. and Tadmor, E. (1990), Non-oscillatory central differencing for hyperbolic conservation laws, J. Comput. Physics 87, pp. 408-463. MR 1047564 (91i:65157)
  • 156. von Neumann, J. (1949), A letter to V. Bush, in ``John von Neumann: selected letters'', Redei M. ed., London Math. Soc. and Americam Math. Soc., History of Mathematics, Vol. 27.
  • 157. Nochetto, R., Siebert K. G. and Veeser A. (2009), Theory of Adaptive Finite Element Methods: an Introduction, Multiscale, Nonlinear and Adaptive Approximation, (R. DeVore and A. Kunoth eds), Springer (2009), 409-542. MR 2648380 (2011k:65164)
  • 158. Oberman, A. (2008), Wide stencil finite difference schemes for the elliptic Monge-Ampere equation and functions of the eigenvalues of the Hessian, DCDS Ser. B 10(1), pp. 221-238. MR 2399429 (2009f:35101)
  • 159. Orszag, S. (1971), Numerical simulation of incompressible flows within simple boundaries. I, Studies in Applied Math. v. 50, pp. 293-327. MR 0305727 (46:4857)
  • 160. Osher, S. J. and Sethian, J. A. (1998), Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79, pp. 12-49. MR 965860 (89h:80012)
  • 161. Osher, S. J. and Fedkiw, R. P. (2003), Level set methods and dynamic implicit surfaces, Appl. Math. Sci, v. 153, Springer. MR 1939127 (2003j:65002)
  • 162. Patera, A. T. (1984), A Spectral Element Method for Fluid Dynamics: Laminar Flow in a Channel Expansion, J.Comput. Phys. v. 54, pp. 468-488.
  • 163. Pedlosky, J. (1987), Geophysical Fluid Dynamics, 2nd ed., Springer.
  • 164. Perelman, G. (2005), Ricci flow with surgery on three-manifolds, preprint, math.DG/0303109.
  • 165. Perona, P. and Malik, J. (1990), Scale-space and edge detection using anisotropic diffusion, Patt. Aanal. Mach. Intell. 12(7), pp. 629-639.
  • 166. Perthame, B. (2007), Transport equations in biology. Frontiers in Mathematics. Birkhäuser Verlag, Basel. MR 2270822 (2007j:35004)
  • 167. Peskin, C. S. (1977), Numerical analysis of blood flow in the heart, J. Comput. Phys. 25 pp. 220-252. MR 0490027 (58:9389)
  • 168. Peskin, C. S. (2002), The immersed boundary method, Acta Numerica, 11, pp. 1-39. MR 2009378 (2004h:74029)
  • 169. Quarteroni, A. (2007), Cardiovascular mathematics. International Congress of Mathematicians. Vol. I, pp. 479-512, Eur. Math. Soc., Zürich. MR 2334201 (2008f:92021)
  • 170. Radó, T. (1933), On the Problem of Plateau, Ergeben. d. Math. u. ihrer Grenzgebiete. Berlin: Springer-Verlag, 1933.
  • 171. Rauch, J. (2012), Hyperbolic Partial Differential Equations and Geometric Optics, Amer. Math. Soc. Graduate Text, v. 133. MR 2918544
  • 172. Raviart, P.-A. (1984), An analysis of particle methods, in Numerical Methods in Fluid Dynamics, vol. 1127 of Lecture Notes in Mathematics, Springer, Berlin, New York, pp. 243-324. MR 802214 (87h:76010)
  • 173. Reed W. H. and Hill, T. R. (1973), Triangular mesh methods for the neutron transport equation, Tech. Report LA-UR-73-479, Los Alamos Sci. Lab.
  • 174. Richardson, L. F. (1922), Weather Prediction by Numerical Process, Cambridge University Press, xii+236 pp. Reprinted by Dover Publications, New York, 1965. MR 2358797 (2008g:86013)
  • 175. Richtmyer, R. and Morton K. W. (1967), Difference methods for initial-value problems, Wiley, reprinted by Krieger Publ. Co., Florida, 1994. MR 1275838 (95b:65003)
  • 176. Rizzo, F. J. (1967), An integral equation approach to boundary value problems of classical elastostatics, Quarterly of Appl. Math., v. 25, pp 83.
  • 177. Roe, P. L. (1981), Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys. v. 43(2), pp. 357-372. MR 640362 (82k:65055)
  • 178. Rudin, L., Osher, S. and Fatemi, E. (1992), Nonlinear total variation based noise removal algorithms, Physica D, 60, 259-268.
  • 179. Saad, Y. (2003), Iterative Methods for Sparse Linear Systems, 2nd edition, SIAM, Philadelphia. MR 1990645 (2004h:65002)
  • 180. Samarskii, A. A. (1965), On monotone difference schemes for elliptic and parabolic equations in the case of a non-selfadjoint elliptic operator, Zh. Vychisl. Mat. i. Mat. Fiz. v. 5, pp. 548-551 (Russian). MR 0189275 (32:6702)
  • 181. Samarskii, A. A. (1971), Introduction to the Theory of Difference Schemes, Nauka, Moscow (Russian). MR 0347102 (49:11822)
  • 182. Serre, D. (1999), Systems of Conservation Laws, Vol. 1&2, Cambridge Univ. Press. MR 1707279 (2000g:35142)
  • 183. Sethian, J. A. (1999), Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision and Materials Science, Cambridge University Press. MR 1700751 (2000c:65015)
  • 184. Sethian, J. A. (1999), Fast Marching Methods, SIAM Review, 41(2), pp. 199-235. MR 1684542 (2000m:65125)
  • 185. Shreve, S. E. (2004), Stochastic calculus for finance. vols I & II. Springer-Verlag, New York.
  • 186. Shu, C.-W. and Osher, S. (1989), Efficient implementation of essentially nonoscillatory shock-capturing schemes. II. J. Comput. Phys. v. 83(1), pp. 32-78. MR 1010162 (90i:65167)
  • 187. Smoller, J. (1994), Shock waves and Reaction-Diffusion Equations, 2nd edition, v. 258, Springer. MR 1301779 (95g:35002)
  • 188. Souganidis. P. (1985), Approximation schemes for viscosity solutions of Hamilton-Jacobi equations, Journal of Differential Equations, v. 57, pp. 1-43. MR 803085 (86k:35028)
  • 189. Strang, G. (1964), Accurate partial difference methods. II. Non-linear problems. Numer. Math. v. 6, pp. 37-46. MR 0166942 (29:4215)
  • 190. Strang, G. and Fix, G. J. (1973), An Analysis of the Finite Element Method, Prentice-Hall, Inc., Englewood Cliffs, NJ. MR 0443377 (56:1747)
  • 191. Struwe, M. (2008), Variational Methods, Springer. MR 2431434 (2009g:49002)
  • 192. Sweby, P. (1984), High-resolution schemes using flux limiters for hyperbolic conservation-laws, SIAM J. Numer. Analysis, v. 21, pp. 995-1011, MR 760628 (85m:65085)
  • 193. Symm, G. T. (1963), Integral equation methods in potential theory. II, Proc. Royal Soc. London A, v. 275(1360), pp. 33-46. MR 0154076 (27:4035)
  • 194. Szabo, B. and Babuška, I. (1991), Finite Element Method Analysis, John Wiley, New-York.
  • 195. Tadmor, E. (1989), Convergence of spectral methods for nonlinear conservation laws, SIAM J. Numer. Anal. Vol. 26, pp. 30-44. MR 977947 (90e:65130)
  • 196. Tadmor, E. (1991), Local error estimates for discontinuous solutions of nonlinear hyperbolic equations, SIAM J. Numer. Anal., Vol. 28 (4), pp. 891-906. MR 1111445 (92d:35190)
  • 197. Tadmor, E. (2002), High resolution methods for time dependent problems with piecewise smooth solutions, in "International Congress of Mathematicians", Proc. the ICM02 Beijing 2002 (Li Tatsien, ed.), Vol. III: Invited lectures, Higher Education Press, pp. 747-757. MR 1957576 (2003m:65150)
  • 198. Tadmor, E. (2003), Entropy stability theory for difference approximations of nonlinear conservation laws and related time dependent problems, Acta Numerica v. 12, pp. 451-512. MR 2249160 (2007g:35150)
  • 199. Tadmor, E. (2007), Filters, mollifiers and the computation of the Gibbs phenomenon, Acta Numerica, v. 16, pp. 305-378. MR 2417931 (2009c:65033)
  • 200. Tadmor, E., Nezzar, S. and Vese, L. (2004), A multiscale image representation using hierarchical (BV,L2) decompositions Multiscale Modeling and Simulations 2(4), pp. 554-579. MR 2113170 (2005h:68163)
  • 201. Tadmor, E. and Zhong, W. (2007), Energy-preserving and stable approximations for the two-dimensional shallow water equations, Proc. the 2006 Abel Symposium, ``Mathematics and Computation, a Contemporary View'', Norway, May, 2006. MR 2503502 (2010e:76090)
  • 202. Tadmor, E. and Tao, T. (2007), Velocity averaging, kinetic formulations and regularizing effects in quasilinear PDEs, Comm. Pure & Applied Math. 60, pp. 1488-1521. MR 2342955 (2008g:35011)
  • 203. Tao, T. (2006), Perelman's proof of the Poincare conjucture: A nonlinear PDE perspective, preprint, arXiv:math.DG0610903. MR 2256586
  • 204. Tao, T. (2006), Nonlinear Dispersive Equations, CBMS Vol. 106, Amer. Math. Soc. MR 2233925 (2008i:35211)
  • 205. Tartar, L. (1979), Compensated compactness and applications to partial differential equations, Research Notes in Math., 39, pp. 136-210. MR 584398 (81m:35014)
  • 206. Tartar, L. (1990), $ H$-measures, a new approach for studying homogenization and concentration effects in partial differential equations. Proc. Roy. Soc. Edinburgh Sect. A, 115 no. 3-4, pp. 193-230. MR 1069518 (91h:35042)
  • 207. Taylor, M. (1996), Partial Differential Equations, Appl. Math. Sci. vol I, II and III, Springer. MR 1395147 (98b:35002a)
  • 208. Temlyakov, V. N. (2006), Greedy approximations, Foundations of Comput. Math. (FoCM), Santander 2005, pp. 371-394, London Math. Soc. Lecture Note Ser., 331, Cambridge Univ. Press, Cambridge. MR 2277112 (2007k:41046)
  • 209. Thomée, V. (1969), Stability theory for partial difference operators, SIAM Rev. 11 pp. 152-195. MR 0250505 (40:3739)
  • 210. Tian, G. (2002), Geometry and nonlinear analysis. Proc. Inter. Congress Math., Vol. I (Beijing, 2002), pp. 475-493, Higher Ed. Press, Beijing2. MR 1989199 (2004j:53052)
  • 211. Tikhonov A. N. and Samarskii, A. A. (1962), Homogeneous difference schemes on nonuniform nets Zh. Vychisl. Mat. i. Mat. Fiz. 2, 812-832 (Russian). MR 0168128 (29:5392)
  • 212. Toro, E. F. (1999), Riemann solvers and numerical methods for fluid dynamics. A practical introduction. Second edition. Springer-Verlag, Berlin. MR 1717819 (2000f:76091)
  • 213. Toselli, A. and Widlund, O. (2004), Domain Decompposition Methods - Algorithms and Theory, Springer Series in Computational Mathematics, v. 34. MR 2104179 (2005g:65006)
  • 214. Trefethen, L. N. (2000), Spectral Methods in Matlab, SIAM. MR 1776072 (2001c:65001)
  • 215. Trefethen, L. N. and Bau, D. (1997), Numerical Linear Algebra, SIAM. MR 1444820 (98k:65002)
  • 216. Turkel, E. (1999), Pre-conditioning Techniques in Computational Fluid Dynamics, Annual Reviews in Fluid Mechanics 1999, V. 31, 385-416. MR 1670946 (99k:76120)
  • 217. Venakides, S. (1990), The Korteweg-de Vries equation with small dispersion: higher order Lax-Levermore theory, Comm. Pure and Appl. Math., v. 43(3) pp. 335-361. MR 1040144 (91k:35236)
  • 218. Villani, C. (2003), Topics in Optimal Transportation, Grad. Course Math., Amer. Math. Soc. Vol. 58. MR 1964483 (2004e:90003)
  • 219. Wesseling, P. (2004), An Introduction to Multigrid Methods, R.T. Edwards Inc. Philadelphia.
  • 220. Xu, J. and Zikatanov, L. (2002), The method of alternating projections and the method of subspace corrections in Hilbert space, J. Amer. Math. Soc. v. 15 pp. 573-597. MR 1896233 (2003f:65095)
  • 221. Zhang, Y.-T., Chen, S., Li, F., Zhao, H. and Shu, C.-W. (2011), Uniformly accurate discontinuous Galerkin fast sweeping methods for Eikonal equations, SIAM J. Scie. Comput., v33(4), pp. 1873-1896. MR 2831038

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Additional Information

Eitan Tadmor
Affiliation: Department of Mathematics and Institute for Physical Science & Technology, Center of Scientific Computation and Mathematical Modeling (CSCAMM), University of Maryland, College Park, Maryland 20742
Email: tadmor@cscamm.umd.edu

DOI: https://doi.org/10.1090/S0273-0979-2012-01379-4
Keywords: Nonlinear PDEs, boundary-value problems, time-dependent problems, well-posed problems, finite-difference methods, finite element methods, finite-volume methods, spectral methods, consistency, accuracy, convergence, stability.
Received by editor(s): March 9, 2011
Received by editor(s) in revised form: May 27, 2012
Published electronically: July 20, 2012
Dedicated: To Heinz-Otto Kreiss with friendship and appreciation
Article copyright: © Copyright 2012 American Mathematical Society

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