Optimal stability polynomials for numerical integration of initial value problems
This work provides a practical tool for maximizing step size in numerical integration of ODEs/PDEs when the spectrum is known, but the method is incremental as it refines existing optimization approaches.
The authors developed a fast, accurate, and robust algorithm based on convex optimization to find optimally stable polynomial approximations to the exponential for one-step integration of initial value problems, achieving the largest stable step size for a given spectrum. The algorithm proves global convergence for first-order approximations and starlike spectra, and demonstrates effectiveness in other cases.
We consider the problem of finding optimally stable polynomial approximations to the exponential for application to one-step integration of initial value ordinary and partial differential equations. The objective is to find the largest stable step size and corresponding method for a given problem when the spectrum of the initial value problem is known. The problem is expressed in terms of a general least deviation feasibility problem. Its solution is obtained by a new fast, accurate, and robust algorithm based on convex optimization techniques. Global convergence of the algorithm is proven in the case that the order of approximation is one and in the case that the spectrum encloses a starlike region. Examples demonstrate the effectiveness of the proposed algorithm even when these conditions are not satisfied.