# American Institute of Mathematical Sciences

June  2012, 17(4): 1229-1259. doi: 10.3934/dcdsb.2012.17.1229

## Jet schemes for advection problems

 1 Temple University, Department of Mathematics, 1805 North Broad Street Philadelphia, PA 19122 2 Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, United States 3 Department of Mathematics and Statistics, McGill University, 805 Sherbrooke W., Montreal, QC, H3A 2K6, Canada

Received  January 2011 Revised  November 2011 Published  February 2012

We present a systematic methodology to develop high order accurate numerical approaches for linear advection problems. These methods are based on evolving parts of the jet of the solution in time, and are thus called jet schemes. Through the tracking of characteristics and the use of suitable Hermite interpolations, high order is achieved in an optimally local fashion, i.e. the update for the data at any grid point uses information from a single grid cell only. We show that jet schemes can be interpreted as advect-and-project processes in function spaces, where the projection step minimizes a stability functional. Furthermore, this function space framework makes it possible to systematically inherit update rules for the higher derivatives from the ODE solver for the characteristics. Jet schemes of orders up to five are applied in numerical benchmark tests, and systematically compared with classical WENO finite difference schemes. It is observed that jet schemes tend to possess a higher accuracy than WENO schemes of the same order.
Citation: Benjamin Seibold, Rodolfo R. Rosales, Jean-Christophe Nave. Jet schemes for advection problems. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1229-1259. doi: 10.3934/dcdsb.2012.17.1229
##### References:
 [1] D. Adalsteinsson and J. A. Sethian, The fast construction of extension velocities in level set methods, J. Comput. Phys., 148 (1999), 2-22. doi: 10.1006/jcph.1998.6090. [2] A. V. Arutyunov, "Optimality Conditions: Abnormal and Degenerate Problems," Mathematics and its Applications, 526, Kluwer Academic Publishers, Dordrecht, 2000. [3] J. B. Bell, P. Colella and H. Glaz, A second-order projection method for the incompressible Navier-Stokes equations, J. Comput. Phys., 85 (1989), 257-283. doi: 10.1016/0021-9991(89)90151-4. [4] M. J. Berger and J. Oliger, Adaptive mesh refinement for hyperbolic partial differential equations, J. Comput. Phys., 53 (1984), 484-512. doi: 10.1016/0021-9991(84)90073-1. [5] J. R. Cash and A. H. Karp, A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides, ACM Trans. Math. Software, 16 (1990), 201-222. doi: 10.1145/79505.79507. [6] P. Chidyagwai, J.-C. Nave, R. R. Rosales and B. Seibold, A comparative study of the efficiency of jet schemes, Int. J. Numer. Anal. Model.-B, under review, 2011, arXiv:1104.0542. [7] B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35 (1998), 2440-2463. doi: 10.1137/S0036142997316712. [8] R. Courant, E. Isaacson and M. Rees, On the solution of nonlinear hyperbolic differential equations by finite differences, Comm. Pure Appl. Math., 5 (1952), 243-255. doi: 10.1002/cpa.3160050303. [9] S. Gottlieb, On high order strong stability preserving Runge-Kutta and multi step time discretizations, J. Sci. Comput., 25 (2005), 105-128. [10] S. Gottlieb, D. I. Ketcheson and C.-W. Shu, High order strong stability preserving time discretizations, Journ. Sci. Computing, 38 (2009), 251-289. doi: 10.1007/s10915-008-9239-z. [11] S. Gottlieb and C.-W. Shu, Total variation diminishing Runge-Kutta schemes, Math. Comp., 67 (1998), 73-85. doi: 10.1090/S0025-5718-98-00913-2. [12] S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev., 43 (2001), 89-112. doi: 10.1137/S003614450036757X. [13] J. P. Heller, An unmixing demonstration, Am. J. Phys., 28 (1960), 348-353. doi: 10.1119/1.1935802. [14] W. Hesthaven and T. Warburton, "Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applicationss," Texts in Applied Mathematics, 54, Springer, New York, 2008. [15] G.-S. Jiang and C.-W. Shu, Efficient implementation of weighted ENO schemes,, J. Comput. Phys., 126 (): 202. [16] M. L. Kontsevich, Lecture at Orsay, December, 1995. [17] R. LeVeque, High-resolution conservative algorithms for advection in incompressible flow, SIAM J. Numer. Anal., 33 (1996), 627-665. doi: 10.1137/0733033. [18] X.-D. Liu, S. Osher and T. Chan, Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115 (1994), 200-212. doi: 10.1006/jcph.1994.1187. [19] C.-H. Min and F. Gibou, A second order accurate level set method on non-graded adaptive cartesian grids, J. Comput. Phys., 225 (2007), 300-321. doi: 10.1016/j.jcp.2006.11.034. [20] J. F. Nash, Jr., Arc structure of singularities, Duke Math. J., 81 (1995), 31-38. doi: 10.1215/S0012-7094-95-08103-4. [21] J.-C. Nave, R. R. Rosales and B. Seibold, A gradient-augmented level set method with an optimally local, coherent advection scheme, J. Comput. Phys., 229 (2010), 3802-3827. doi: 10.1016/j.jcp.2010.01.029. [22] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2. [23] W. H. Reed and T. R. Hill, Triangular mesh methods for the neutron transport equation, Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973. [24] C.-W. Shu, Total-variation-diminishing time discretizations, SIAM J. Sci. Statist. Comput., 9 (1988), 1073-1084. doi: 10.1137/0909073. [25] C.-W. Shu and S. Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes, J. Comput. Phys., 77 (1988), 439-471. doi: 10.1016/0021-9991(88)90177-5. [26] M. Sussman, P. Smereka and S. Osher, A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys., 114 (1994), 146-159. doi: 10.1006/jcph.1994.1155. [27] G. I. Taylor, "Low Reynolds Number Flow," Movie, U.S. National Committee for Fluid Mechanics Films (NCFMF), 1961. [28] B. van Leer, Towards the ultimate conservative difference scheme I. The quest of monoticity, Springer Lecture Notes in Physics, 18 (1973), 163-168. doi: 10.1007/BFb0118673. [29] B. van Leer, Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method, J. Comput. Phys., 32 (1979), 101-136. doi: 10.1016/0021-9991(79)90145-1.

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##### References:
 [1] D. Adalsteinsson and J. A. Sethian, The fast construction of extension velocities in level set methods, J. Comput. Phys., 148 (1999), 2-22. doi: 10.1006/jcph.1998.6090. [2] A. V. Arutyunov, "Optimality Conditions: Abnormal and Degenerate Problems," Mathematics and its Applications, 526, Kluwer Academic Publishers, Dordrecht, 2000. [3] J. B. Bell, P. Colella and H. Glaz, A second-order projection method for the incompressible Navier-Stokes equations, J. Comput. Phys., 85 (1989), 257-283. doi: 10.1016/0021-9991(89)90151-4. [4] M. J. Berger and J. Oliger, Adaptive mesh refinement for hyperbolic partial differential equations, J. Comput. Phys., 53 (1984), 484-512. doi: 10.1016/0021-9991(84)90073-1. [5] J. R. Cash and A. H. Karp, A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides, ACM Trans. Math. Software, 16 (1990), 201-222. doi: 10.1145/79505.79507. [6] P. Chidyagwai, J.-C. Nave, R. R. Rosales and B. Seibold, A comparative study of the efficiency of jet schemes, Int. J. Numer. Anal. Model.-B, under review, 2011, arXiv:1104.0542. [7] B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35 (1998), 2440-2463. doi: 10.1137/S0036142997316712. [8] R. Courant, E. Isaacson and M. Rees, On the solution of nonlinear hyperbolic differential equations by finite differences, Comm. Pure Appl. Math., 5 (1952), 243-255. doi: 10.1002/cpa.3160050303. [9] S. Gottlieb, On high order strong stability preserving Runge-Kutta and multi step time discretizations, J. Sci. Comput., 25 (2005), 105-128. [10] S. Gottlieb, D. I. Ketcheson and C.-W. Shu, High order strong stability preserving time discretizations, Journ. Sci. Computing, 38 (2009), 251-289. doi: 10.1007/s10915-008-9239-z. [11] S. Gottlieb and C.-W. Shu, Total variation diminishing Runge-Kutta schemes, Math. Comp., 67 (1998), 73-85. doi: 10.1090/S0025-5718-98-00913-2. [12] S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev., 43 (2001), 89-112. doi: 10.1137/S003614450036757X. [13] J. P. Heller, An unmixing demonstration, Am. J. Phys., 28 (1960), 348-353. doi: 10.1119/1.1935802. [14] W. Hesthaven and T. Warburton, "Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applicationss," Texts in Applied Mathematics, 54, Springer, New York, 2008. [15] G.-S. Jiang and C.-W. Shu, Efficient implementation of weighted ENO schemes,, J. Comput. Phys., 126 (): 202. [16] M. L. Kontsevich, Lecture at Orsay, December, 1995. [17] R. LeVeque, High-resolution conservative algorithms for advection in incompressible flow, SIAM J. Numer. Anal., 33 (1996), 627-665. doi: 10.1137/0733033. [18] X.-D. Liu, S. Osher and T. Chan, Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115 (1994), 200-212. doi: 10.1006/jcph.1994.1187. [19] C.-H. Min and F. Gibou, A second order accurate level set method on non-graded adaptive cartesian grids, J. Comput. Phys., 225 (2007), 300-321. doi: 10.1016/j.jcp.2006.11.034. [20] J. F. Nash, Jr., Arc structure of singularities, Duke Math. J., 81 (1995), 31-38. doi: 10.1215/S0012-7094-95-08103-4. [21] J.-C. Nave, R. R. Rosales and B. Seibold, A gradient-augmented level set method with an optimally local, coherent advection scheme, J. Comput. Phys., 229 (2010), 3802-3827. doi: 10.1016/j.jcp.2010.01.029. [22] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2. [23] W. H. Reed and T. R. Hill, Triangular mesh methods for the neutron transport equation, Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973. [24] C.-W. Shu, Total-variation-diminishing time discretizations, SIAM J. Sci. Statist. Comput., 9 (1988), 1073-1084. doi: 10.1137/0909073. [25] C.-W. Shu and S. Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes, J. Comput. Phys., 77 (1988), 439-471. doi: 10.1016/0021-9991(88)90177-5. [26] M. Sussman, P. Smereka and S. Osher, A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys., 114 (1994), 146-159. doi: 10.1006/jcph.1994.1155. [27] G. I. Taylor, "Low Reynolds Number Flow," Movie, U.S. National Committee for Fluid Mechanics Films (NCFMF), 1961. [28] B. van Leer, Towards the ultimate conservative difference scheme I. The quest of monoticity, Springer Lecture Notes in Physics, 18 (1973), 163-168. doi: 10.1007/BFb0118673. [29] B. van Leer, Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method, J. Comput. Phys., 32 (1979), 101-136. doi: 10.1016/0021-9991(79)90145-1.
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