Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  new three-stage W-methods for the time integration of semi-discretized advection diffusion reaction Partial Differential Equations (PDEs) are provided. In particular, two three-parametric families of W-methods of order three are obtained under a realistic assumption regarding the commutator of the exact Jacobian and the approximation of the Jacobian which defines the corresponding W-method. Specific methods are selected by minimizing error coefficients, enlarging stability regions or increasing monotonicity factors, and embedded methods of order two for an adaptive time integration are derived by further assuming first order approximations to the Jacobian. The relevance of the newly proposed methods in connection with the Approximate Matrix Factorization technique is discussed and numerical illustration on practical PDE problems revealing that the new methods are good competitors over existing integrators in the literature is provided.
['S. González-Pinto', 'D. Hernández-Abreu', 'Severiano Pérez-Rodríguez', 'Rüdiger Weiner']
I am writing a research paper on a idea A family of three-stage third order AMF-W-methods for the time integration of advection diffusion reaction PDEs.
[{"from": "human", "value": "I am writing a research paper on a idea A family of three-stage third order AMF-W-methods for the time integration of advection diffusion reaction PDEs."}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  new three-stage W-methods for the time integration of semi-discretized advection diffusion reaction Partial Differential Equations (PDEs) are provided. In particular, two three-parametric families of W-methods of order three are obtained under a realistic assumption regarding the commutator of the exact Jacobian and the approximation of the Jacobian which defines the corresponding W-method. Specific methods are selected by minimizing error coefficients, enlarging stability regions or increasing monotonicity factors, and embedded methods of order two for an adaptive time integration are derived by further assuming first order approximations to the Jacobian. The relevance of the newly proposed methods in connection with the Approximate Matrix Factorization technique is discussed and numerical illustration on practical PDE problems revealing that the new methods are good competitors over existing integrators in the literature is provided."}]