A higher order frozen Jacobian iterative method for solving HamiltonJacobi equations
Authors
Ebraheem O. Alzahrani
 Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia.
Eman S. AlAidarous
 Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia.
Arshad M. M. Younas
 Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia.
Fayyaz Ahmad
 Dipartimento di Scienza e Alta Tecnologia, Universita dell'Insubria, Via Valleggio 11, Como 22100, Italy.
 ment de Fisica i Enginyeria Nuclear, Universitat Politecnica de Catalunya, Comte d'Urgell 187, 08036 Barcelona, Spain.
 UCERD Islamabad, Pakistan.
Shamshad Ahmad
 Heat and Mass Transfer Technological Center, Technical University of Catalonia, Colom 11, 08222 Terrassa, Spain.
Shahid Ahmad
 Department of Mathematics, Government College University Lahore, Lahore, Pakistan.
Abstract
It is wellknown that the solution of HamiltonJacobi equation may have singularity i.e., the solution is
nonsmooth or nearly nonsmooth. We construct a frozen Jacobian multistep iterative method for solving
HamiltonJacobi equation under the assumption that the solution is nearly singular. The frozen Jacobian
iterative methods are computationally very efficient because a single instance of the iterative method uses a
single inversion (in the scene of LU factorization) of the frozen Jacobian. The multistep part enhances the
convergence order by solving lower and upper triangular systems. The convergence order of our proposed
iterative method is \(3(m  1)\) for \(m \geq 3\). For attaining good numerical accuracy in the solution, we use
Chebyshev pseudospectral collocation method. Some HamiltonJacobi equations are solved, and numerically
obtained results show high accuracy.
Share and Cite
ISRP Style
Ebraheem O. Alzahrani, Eman S. AlAidarous, Arshad M. M. Younas, Fayyaz Ahmad, Shamshad Ahmad, Shahid Ahmad, A higher order frozen Jacobian iterative method for solving HamiltonJacobi equations, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 12, 62106227
AMA Style
Alzahrani Ebraheem O., AlAidarous Eman S., Younas Arshad M. M., Ahmad Fayyaz, Ahmad Shamshad, Ahmad Shahid, A higher order frozen Jacobian iterative method for solving HamiltonJacobi equations. J. Nonlinear Sci. Appl. (2016); 9(12):62106227
Chicago/Turabian Style
Alzahrani, Ebraheem O., AlAidarous, Eman S., Younas, Arshad M. M., Ahmad, Fayyaz, Ahmad, Shamshad, Ahmad, Shahid. "A higher order frozen Jacobian iterative method for solving HamiltonJacobi equations." Journal of Nonlinear Sciences and Applications, 9, no. 12 (2016): 62106227
Keywords
 HamiltonJacobi equations
 frozen Jacobian iterative methods
 systems of nonlinear equations
 Chebyshev pseudospectral collocation method.
MSC
References

[1]
R. Abgrall, Numerical discretization of the firstorder HamiltonJacobi equation on triangular meshes, Comm. Pure Appl. Math., 49 (1996), 13391373

[2]
F. Ahmad, E. Tohidi, J. A. Carrasco, A parameterized multistep Newton method for solving systems of nonlinear equations, Numer. Algorithms, 71 (2016), 631653

[3]
F. Ahmad, E. Tohidi, M. Z. Ullah, J. A. Carrasco, Higher order multistep Jarrattlike method for solving systems of nonlinear equations: application to PDEs and ODEs, Comput. Math. Appl., 70 (2015), 624636

[4]
E. S. AlAidarous, E. O. Alzahrani, H. Ishii, A. M. M. Younas, Asymptotic analysis for the eikonal equation with the dynamical boundary conditions, Math. Nachr., 287 (2014), 15631588

[5]
E. S. AlAidarous, E. O. Alzahrani, H. Ishii, A. M. M. Younas, A convergence result for the ergodic problem for HamiltonJacobi equations with Neumanntype boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 225242

[6]
E. S. Alaidarous, M. Z. Ullah, F. Ahmad, A. S. AlFhaid, An efficient higherorder quasilinearization method for solving nonlinear BVPs, J. Appl. Math., 2013 (2013), 11 pages

[7]
A. H. Bhrawy, E. H. Doha, M. A. Abdelkawy, R. A. Van Gorder, JacobiGaussLobatto collocation method for solving nonlinear reactiondiffusion equations subject to Dirichlet boundary conditions, Appl. Math. Model., 40 (2016), 17031716

[8]
M. G. Crandall, H. Ishii, P.L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 167

[9]
M. G. Crandall, P.L. Lions, Viscosity solutions of HamiltonJacobi equations, Trans. Amer. Math. Soc., 277 (1983), 142

[10]
M. Dehghan, F. FakharIzadi, The spectral collocation method with three different bases for solving a nonlinear partial differential equation arising in modeling of nonlinear waves, Math. Comput. Modelling, 53 (2011), 18651877

[11]
E. H. Doha, A. H. Bhrawy, M. A. Abdelkawy, R. A. Van Gorder, JacobiGaussLobatto collocation method for the numerical solution of 1 + 1 nonlinear Schrödinger equations, J. Comput. Phys., 261 (2014), 244255

[12]
E. H. Doha, A. H. Bhrawy, S. S. EzzEldien, Efficient Chebyshev spectral methods for solving multiterm fractional orders differential equations, Appl. Math. Model., 35 (2011), 56625672

[13]
A. Harten, B. Engquist, S. Osher, S. R. Chakravarthy, Uniformly high order accurate essentially nonoscillatory schemes, III, J. Comput. Phys., 131 (1997), 347

[14]
G.S. Jiang, D.P. Peng, Weighted ENO schemes for HamiltonJacobi equations, SIAM J. Sci. Comput., 21 (2000), 21262143

[15]
G.S. Jiang, C.W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126 (1996), 202228

[16]
P.L. Lions, Generalized solutions of HamiltonJacobi equations, Research Notes in Mathematics, Pitman (Advanced Publishing Program), Boston, Mass.London (1982)

[17]
P.L. Lions, P. E. Souganidis, Convergence of MUSCL and filtered schemes for scalar conservation laws and HamiltonJacobi equations, Numer. Math., 69 (1995), 441470

[18]
X.D. Liu, S. Osher, T. Chan, Weighted essentially nonoscillatory schemes, J. Comput. Phys., 115 (1994), 200212

[19]
H. Montazeri, F. Soleymani, S. Shateyi, S. S. Motsa, On a new method for computing the numerical solution of systems of nonlinear equations, J. Appl. Math., 2012 (2012), 15 pages

[20]
J. M. Ortega, W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York (1970)

[21]
S. Osher, J. A. Sethian, Fronts propagating with curvaturedependent speed: algorithms based on HamiltonJacobi formulations, J. Comput. Phys., 79 (1988), 1249

[22]
S. Osher, C.W. Shu, Highorder essentially nonoscillatory schemes for HamiltonJacobi equations, SIAM J. Numer. Anal., 28 (1991), 907922

[23]
U. Qasim, Z. Ali, F. Ahmad, S. SerraCapizzano, M. Z. Ullah, M. Asma, Constructing frozen Jacobian iterative methods for solving systems of nonlinear equations, associated with ODEs and PDEs using the homotopy method,Constructing frozen Jacobian iterative methods for solving systems of nonlinear equations, associated with ODEs and PDEs using the homotopy method,Constructing frozen Jacobian iterative methods for solving systems of nonlinear equations, associated with ODEs and PDEs using the homotopy method,Constructing frozen Jacobian iterative methods for solving systems of nonlinear equations, associated with ODEs and PDEs using the homotopy method,Constructing frozen Jacobian iterative methods for solving systems of nonlinear equations, associated with ODEs and PDEs using the homotopy method,Constructing frozen Jacobian iterative methods for solving systems of nonlinear equations, associated with ODEs and PDEs using the homotopy method, Algorithms (Basel), 9 (2016), 17 pages

[24]
S. Qasim, Z. Ali, F. Ahmad, S. SerraCapizzano, M. Z. Ullah, A. Mahmood, Solving systems of nonlinear equations when the nonlinearity is expensive, Comput. Math. Appl., 71 (2016), 14641478

[25]
J. Shen, T. Tang, L.L. Wang, Spectral methods: algorithms, analysis and applications, Springer Series in Computational Mathematics, Springer, (2011)

[26]
P. E. Souganidis, Approximation schemes for viscosity solutions of HamiltonJacobi equations, J. Differential Equations, 59 (1985), 143

[27]
G. Szegö, Orthogonal Polynomials, American Mathematical Society Colloquium Publications, American Mathematical Society, New York (1939)

[28]
J. F. Traub, Itersative methods for the solution of equations, PrenticeHall, Englewood Cliffs (1964)

[29]
E. Tohidi, S. Lotfi Noghabi, An efficient Legendre pseudospectral method for solving nonlinear quasi bangbang optimal control problems, J. Appl. Math. Stat. Inform., 8 (2012), 7385

[30]
M. Z. Ullah, S. SerraCapizzano, F. Ahmad, An efficient multistep iterative method for computing the numerical solution of systems of nonlinear equations associated with ODEs, Appl. Math. Comput., 250 (2015), 249259

[31]
M. Z. Ullah, F. Soleymani, A. S. AlFhaid, Numerical solution of nonlinear systems by a general class of iterative methods with application to nonlinear PDEs, Numer. Algorithms, 67 (2014), 223242