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.
Keywords
 HamiltonJacobi equations
 frozen Jacobian iterative methods
 systems of nonlinear equations
 Chebyshev pseudospectral collocation method.
MSC
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