# A higher order frozen Jacobian iterative method for solving Hamilton-Jacobi equations

Volume 9, Issue 12, pp 6210--6227
• 1825 Views ### Authors

Ebraheem O. Alzahrani - Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia. Eman S. Al-Aidarous - 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 well-known that the solution of Hamilton-Jacobi equation may have singularity i.e., the solution is non-smooth or nearly non-smooth. We construct a frozen Jacobian multi-step iterative method for solving Hamilton-Jacobi 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 multi-step 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 pseudo-spectral collocation method. Some Hamilton-Jacobi equations are solved, and numerically obtained results show high accuracy.

### Share and Cite

##### ISRP Style

Ebraheem O. Alzahrani, Eman S. Al-Aidarous, Arshad M. M. Younas, Fayyaz Ahmad, Shamshad Ahmad, Shahid Ahmad, A higher order frozen Jacobian iterative method for solving Hamilton-Jacobi equations, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 12, 6210--6227

##### AMA Style

Alzahrani Ebraheem O., Al-Aidarous Eman S., Younas Arshad M. M., Ahmad Fayyaz, Ahmad Shamshad, Ahmad Shahid, A higher order frozen Jacobian iterative method for solving Hamilton-Jacobi equations. J. Nonlinear Sci. Appl. (2016); 9(12):6210--6227

##### Chicago/Turabian Style

Alzahrani, Ebraheem O., Al-Aidarous, Eman S., Younas, Arshad M. M., Ahmad, Fayyaz, Ahmad, Shamshad, Ahmad, Shahid. "A higher order frozen Jacobian iterative method for solving Hamilton-Jacobi equations." Journal of Nonlinear Sciences and Applications, 9, no. 12 (2016): 6210--6227

### Keywords

• Hamilton-Jacobi equations
• frozen Jacobian iterative methods
• systems of nonlinear equations
• Chebyshev pseudo-spectral collocation method.

•  65M70
•  35F21
•  65M12

### References

•  R. Abgrall, Numerical discretization of the first-order Hamilton-Jacobi equation on triangular meshes, Comm. Pure Appl. Math., 49 (1996), 1339--1373

•  F. Ahmad, E. Tohidi, J. A. Carrasco, A parameterized multi-step Newton method for solving systems of nonlinear equations, Numer. Algorithms, 71 (2016), 631--653

•  F. Ahmad, E. Tohidi, M. Z. Ullah, J. A. Carrasco, Higher order multi-step Jarratt-like method for solving systems of nonlinear equations: application to PDEs and ODEs, Comput. Math. Appl., 70 (2015), 624--636

•  E. S. Al-Aidarous, E. O. Alzahrani, H. Ishii, A. M. M. Younas, Asymptotic analysis for the eikonal equation with the dynamical boundary conditions, Math. Nachr., 287 (2014), 1563--1588

•  E. S. Al-Aidarous, E. O. Alzahrani, H. Ishii, A. M. M. Younas, A convergence result for the ergodic problem for Hamilton-Jacobi equations with Neumann-type boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 225--242

•  E. S. Alaidarous, M. Z. Ullah, F. Ahmad, A. S. Al-Fhaid, An efficient higher-order quasilinearization method for solving nonlinear BVPs, J. Appl. Math., 2013 (2013), 11 pages

•  A. H. Bhrawy, E. H. Doha, M. A. Abdelkawy, R. A. Van Gorder, Jacobi-Gauss-Lobatto collocation method for solving nonlinear reaction-diffusion equations subject to Dirichlet boundary conditions, Appl. Math. Model., 40 (2016), 1703--1716

•  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), 1--67

•  M. G. Crandall, P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1--42

•  M. Dehghan, F. Fakhar-Izadi, 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), 1865--1877

•  E. H. Doha, A. H. Bhrawy, M. A. Abdelkawy, R. A. Van Gorder, Jacobi-Gauss-Lobatto collocation method for the numerical solution of 1 + 1 nonlinear Schrödinger equations, J. Comput. Phys., 261 (2014), 244--255

•  E. H. Doha, A. H. Bhrawy, S. S. Ezz-Eldien, Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations, Appl. Math. Model., 35 (2011), 5662--5672

•  A. Harten, B. Engquist, S. Osher, S. R. Chakravarthy, Uniformly high order accurate essentially non-oscillatory schemes, III, J. Comput. Phys., 131 (1997), 3--47

•  G.-S. Jiang, D.-P. Peng, Weighted ENO schemes for Hamilton-Jacobi equations, SIAM J. Sci. Comput., 21 (2000), 2126--2143

•  G.-S. Jiang, C.-W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126 (1996), 202--228

•  P.-L. Lions, Generalized solutions of Hamilton-Jacobi equations, Research Notes in Mathematics, Pitman (Advanced Publishing Program), Boston, Mass.-London (1982)

•  P.-L. Lions, P. E. Souganidis, Convergence of MUSCL and filtered schemes for scalar conservation laws and Hamilton-Jacobi equations, Numer. Math., 69 (1995), 441--470

•  X.-D. Liu, S. Osher, T. Chan, Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115 (1994), 200--212

•  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

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

•  S. Osher, J. A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12--49

•  S. Osher, C.-W. Shu, High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations, SIAM J. Numer. Anal., 28 (1991), 907--922

•  U. Qasim, Z. Ali, F. Ahmad, S. Serra-Capizzano, 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

•  S. Qasim, Z. Ali, F. Ahmad, S. Serra-Capizzano, M. Z. Ullah, A. Mahmood, Solving systems of nonlinear equations when the nonlinearity is expensive, Comput. Math. Appl., 71 (2016), 1464--1478

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

•  P. E. Souganidis, Approximation schemes for viscosity solutions of Hamilton-Jacobi equations, J. Differential Equations, 59 (1985), 1--43

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

•  J. F. Traub, Itersative methods for the solution of equations, Prentice-Hall, Englewood Cliffs (1964)

•  E. Tohidi, S. Lotfi Noghabi, An efficient Legendre pseudospectral method for solving nonlinear quasi bang-bang optimal control problems, J. Appl. Math. Stat. Inform., 8 (2012), 73--85

•  M. Z. Ullah, S. Serra-Capizzano, F. Ahmad, An efficient multi-step iterative method for computing the numerical solution of systems of nonlinear equations associated with ODEs, Appl. Math. Comput., 250 (2015), 249--259

•  M. Z. Ullah, F. Soleymani, A. S. Al-Fhaid, Numerical solution of nonlinear systems by a general class of iterative methods with application to nonlinear PDEs, Numer. Algorithms, 67 (2014), 223--242