For the accuracy and stability of numerical codes, we have to ensure that a discretisation is consistent and is stable. Both depend on the truncation error, while we distinguish zero- and A-stability. We end up with the notion of convergence according to the Lax Equivalence Theorem, and finally discuss how we can compute the convergence order experimentally.
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I call \(\lambda \) the material parameter. Different to flow through a subsurface medium, e.g., our \(\lambda \) is not really a material. But the name material here highlights that it is not a parameter determined by yet another equation but something fixed.
Some maths books define Lipschitz-continuity “simply” as \(|F(s_1)-F(s_2)| \le C |s_1-s_2|\). In our discussion, we split up this s into \(s=(t,f(t))\) as we are interested in ODEs, and we wobble around with the f(t) part only. The more general definition from math books shows that we also can slightly alter the t argument. The solution will not change too much either. Both definitions focus on the right-hand side of the ODE. As the right side determines the solution, its (continuity) properties carry over to the solution.