Linear iterative solvers for implicit ODE methods

The numerical solution of stiff initial value problems, which lead to the problem of solving large systems of mildly nonlinear equations are considered. For many problems derived from engineering and science, a solution is possible only with methods derived from iterative linear equation solvers. A common approach to solving the nonlinear equations is to employ an approximate solution obtained from an explicit method. The error is examined to determine how it is distributed among the stiff and non-stiff components, which bears on the choice of an iterative method. The conclusion is that error is (roughly) uniformly distributed, a fact that suggests the Chebyshev method (and the accompanying Manteuffel adaptive parameter algorithm). This method is described, also commenting on Richardson's method and its advantages for large problems. Richardson's method and the Chebyshev method with the Mantueffel algorithm are applied to the solution of the nonlinear equations by Newton's method

Canonical Runge—Kutta—Nyström methods of orders five and six

AbstractIn this paper, we construct canonical explicit five-stage and seven-stage Runge—Kutta—Nyström methods of orders five and six, respectively, for Hamiltonian dynamical systems

G protein-coupled receptor-mediated calcium signaling in astrocytes

Astrocytes express a large variety of G~protein-coupled receptors (GPCRs) which mediate the transduction of extracellular signals into intracellular calcium responses. This transduction is provided by a complex network of biochemical reactions which mobilizes a wealth of possible calcium-mobilizing second messenger molecules. Inositol 1,4,5-trisphosphate is probably the best known of these molecules whose enzymes for its production and degradation are nonetheless calcium-dependent. We present a biophysical modeling approach based on the assumption of Michaelis-Menten enzyme kinetics, to effectively describe GPCR-mediated astrocytic calcium signals. Our model is then used to study different mechanisms at play in stimulus encoding by shape and frequency of calcium oscillations in astrocytes.Comment: 35 pages, 6 figures, 1 table, 3 appendices (book chapter

Comments on “A critical appraisal of Markov state models” by M. Sarich and C. Schütte

The article by Sarich and Schütte takes a broad view of the use of Markov state models, with biomolecules as the intended application, and proposes novel approaches based on mathematical insights