Pursuing of Adaptive Linear Multistep Methods

A Hungarian-Swedish cooperation

At the end of January two renowned guest professors joined Department of Applied Analysis and Computational Mathematics at Eötvös Loránd University. Professors Carmen Arévalo (Lund University, Sweden) and Gustaf Söderlind (Lund University, Sweden) had an enormous influence to the numerical analysis group at the Department. Professor Söderlind also served as the Distinguished Guest Scientist’s Programme winner of the Hungarian Academy of Sciences.

During the last half year the group has been mainly focusing on adaptive linear multistep methods (LMMs) related projects. Adaptive time-stepping is a central issue for the efficient solution of initial value problems (IVPs) in ODEs. During the 90’s and the beginning of the Millennium Professor Söderlind and his former PhD student Professor Kjell Gustafsson introduced a new framework to numerical analysts solving efficiently IVPs by adaptive Runge–Kutta methods. Their seminal papers were based on techniques from control theory that have proved efficient in the construction of adaptive ODE software. The methodology rested on classical theories of linear difference equations, difference operators and stability.

In case of LMMs numerical analysts didn’t have efficient adaptive methods. However, due to the 2017 paper entitled Grid-independent Constraction of Multistep Methods by Professors Arévalo and Söderlind variable step size LMMs have a new polynomial formulation. This representation opened the door pursuing efficient LMMs.

Using this formulation and Toeplitz operator techniqe we could prove the 0-stability of these methods on smooth nonuniform grids. Basically our 2018 results mean that a strongly stable constant step size LMM is 0-stable on every smooth nonuniform grid. After the professors arrival we have initiated LMMs related projects.

First, we have started to extend our technique to the full problem. The project has been started in January and we have promising results. We have to overcome some techniqual difficulties showing the full convergence of adaptive LMMs for IVPs in ODEs.

Second, we wanted to show that our theoretical results fit into the computational practice. It turned out that in variable step size implementations the error model must be dynamic and include past step ratios, even in the asymptotic regime. We derived dynamic asymptotic models of the local error and its estimator, and showed how to use dynamically compensated step size controllers that keep the asymptotic local error near a prescribed tolerance tol. The new error models enabled the use of controllers with enhanced stability, producing more regular step size sequences.

Our fruitful collaboration led to continue joint research projects in this field and initiate a co-supervised PhD project. The latter one is going to focus on further developments of a new standard and a next generation technique in multistep software.

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