Generalize the error propagation matrix to ODE systems

Investigate whether the error propagation matrix introduced for the Van der Pol equation can be developed and applied to systems of ordinary differential equations in general.

Background

The lecture notes introduce an error propagation matrix for the Van der Pol system in Sections 3.3 and 3.4. This matrix approximates the propagation of the global error from one time step to the next while retaining multidimensional information that is lost in scalar Gronwall-type estimates.

The notes identify as an unresolved topic whether this construction extends beyond the Van der Pol example to general systems of ordinary differential equations. Such an extension would support more informative long-time error estimates for nonlinear ODE systems with directional growth, contraction, and rotation.

References

A few examples of open topics are listed below. For ODEs, one topic worth to be studied is the error propagation matrix which was introduced in Section 3.3 and 3.4 for the case study of the Van der Pol equation. This kind of error propagation matrix might work for systems of ODEs in general.

— Numerical Smoothness and Error Analysis for Evolution Equations  (2610.12888 - Sun, 9 Oct 2026) in Section 4, “Future research”