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

Volume 9, Issue 12, pp 6210--6227 Publication Date: December 27, 2016
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### 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.

### Keywords

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

•  65M70
•  35F21
•  65M12

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