Numerical analysis of solvers for non-linear ordinary differential equations

The goal for the project is to analyze and test a family of solvers for ordinary differential equations (ODEs) known as generalized Rush-Larsen (GRL) methods.
Master

The electrical activity in the heart can be described by a coupled system of partial and ordinary differential equations (PDEs and ODEs). The PDEs describe electro-diffusive properties in the tissue, while the ODEs describe the electrophysiology of individual cells. The coupled model is extremely challenging to solve, and substantial research efforts have been put into efficient numerical methods for all model components. For the cell model ODEs, a popular group of methods have been developed based on a local linearization of the generally non-linear ODEs, an idea originally presented by Rush and Larsen [1]. Even today the method is widely used in the research community, and several extensions have been derived that increase the accuracy and applicability of the method, see for instance [2]. These methods, often referred to as Generalized Rush-Larsen (GRL) methods, work remarkably well in practice, but their numerical properties are not fully understood. In this project we will analyze the accuracy and stability of a family of GRL methods, to derive theoretical bounds and improved insight into the applicability of the methods. This project is suitable for a student with interests in numerical computing and in analysis of numerical methods.

Goal

The goal is to analyze a family of ODE solvers of the GRL class, to establish theoretical estimates on stability and accuracy of the methods. The theoretical results will be tested in numerical experiments.

Learning outcome

The student will gain experience in numerical analysis, computational methods for nonlinear ODEs, and modeling of physiological systems.

Qualifications

• Fundamental knowledge of numerical analysis
• Basic understanding of ordinary differential equations
• Interest in numerical analysis and computing

Supervisors

  • Joakim Sundnes
  • Aslak Tveito

References

[1] S. Rush, H. Larsen, A practical algorithm for solving dynamic membrane equations, IEEE Trans. Biomed. Eng., vol. BME-25, no. 4, 1978.
[2] J. Sundnes, R. Artebrant, O. Skavhaug, A. Tveito. A second-order algorithm for solving dynamic cell membrane equations, IEEE Trans. Biomed. Eng 56 (10), 2009.

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