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.

show all references

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.

[1]

Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5495-5508. doi: 10.3934/dcdsb.2020355

[2]

Marc Wolff, Stéphane Jaouen, Hervé Jourdren, Eric Sonnendrücker. High-order dimensionally split Lagrange-remap schemes for ideal magnetohydrodynamics. Discrete and Continuous Dynamical Systems - S, 2012, 5 (2) : 345-367. doi: 10.3934/dcdss.2012.5.345

[3]

Kai Jiang, Wei Si. High-order energy stable schemes of incommensurate phase-field crystal model. Electronic Research Archive, 2020, 28 (2) : 1077-1093. doi: 10.3934/era.2020059

[4]

Zheng Sun, José A. Carrillo, Chi-Wang Shu. An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems. Kinetic and Related Models, 2019, 12 (4) : 885-908. doi: 10.3934/krm.2019033

[5]

Amy Allwright, Abdon Atangana. Augmented upwind numerical schemes for a fractional advection-dispersion equation in fractured groundwater systems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 443-466. doi: 10.3934/dcdss.2020025

[6]

Lela Dorel. Glucose level regulation via integral high-order sliding modes. Mathematical Biosciences & Engineering, 2011, 8 (2) : 549-560. doi: 10.3934/mbe.2011.8.549

[7]

Guoshan Zhang, Peizhao Yu. Lyapunov method for stability of descriptor second-order and high-order systems. Journal of Industrial and Management Optimization, 2018, 14 (2) : 673-686. doi: 10.3934/jimo.2017068

[8]

Yuezheng Gong, Jiaquan Gao, Yushun Wang. High order Gauss-Seidel schemes for charged particle dynamics. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 573-585. doi: 10.3934/dcdsb.2018034

[9]

Abdelwahab Bensouilah, Sahbi Keraani. Smoothing property for the $ L^2 $-critical high-order NLS Ⅱ. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2961-2976. doi: 10.3934/dcds.2019123

[10]

Kolade M. Owolabi, Abdon Atangana. High-order solvers for space-fractional differential equations with Riesz derivative. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 567-590. doi: 10.3934/dcdss.2019037

[11]

Marc Bonnet. Inverse acoustic scattering using high-order small-inclusion expansion of misfit function. Inverse Problems and Imaging, 2018, 12 (4) : 921-953. doi: 10.3934/ipi.2018039

[12]

Raymond H. Chan, Haixia Liang, Suhua Wei, Mila Nikolova, Xue-Cheng Tai. High-order total variation regularization approach for axially symmetric object tomography from a single radiograph. Inverse Problems and Imaging, 2015, 9 (1) : 55-77. doi: 10.3934/ipi.2015.9.55

[13]

Shan Jiang, Li Liang, Meiling Sun, Fang Su. Uniform high-order convergence of multiscale finite element computation on a graded recursion for singular perturbation. Electronic Research Archive, 2020, 28 (2) : 935-949. doi: 10.3934/era.2020049

[14]

Tarek Saanouni. Global well-posedness of some high-order semilinear wave and Schrödinger type equations with exponential nonlinearity. Communications on Pure and Applied Analysis, 2014, 13 (1) : 273-291. doi: 10.3934/cpaa.2014.13.273

[15]

Phillip Colella. High-order finite-volume methods on locally-structured grids. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4247-4270. doi: 10.3934/dcds.2016.36.4247

[16]

Andrey B. Muravnik. On the Cauchy problem for differential-difference parabolic equations with high-order nonlocal terms of general kind. Discrete and Continuous Dynamical Systems, 2006, 16 (3) : 541-561. doi: 10.3934/dcds.2006.16.541

[17]

Lixuan Zhang, Xuefei Yang. On pole assignment of high-order discrete-time linear systems with multiple state and input delays. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022022

[18]

Xuefeng Shen, Khoa Tran, Melvin Leok. High-order symplectic Lie group methods on $ SO(n) $ using the polar decomposition. Journal of Computational Dynamics, 2022  doi: 10.3934/jcd.2022003

[19]

Roger P. de Moura, Ailton C. Nascimento, Gleison N. Santos. On the stabilization for the high-order Kadomtsev-Petviashvili and the Zakharov-Kuznetsov equations with localized damping. Evolution Equations and Control Theory, 2022, 11 (3) : 711-727. doi: 10.3934/eect.2021022

[20]

Anis Theljani, Ke Chen. An augmented lagrangian method for solving a new variational model based on gradients similarity measures and high order regulariation for multimodality registration. Inverse Problems and Imaging, 2019, 13 (2) : 309-335. doi: 10.3934/ipi.2019016

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (90)
  • HTML views (0)
  • Cited by (12)

[Back to Top]