Talk:Initial value problems

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    Review of “Initial value Problems” by L.F. Shampine and Skip Thompson

    This article concentrates on numerical approaches to solving initial value problems in ordinary differential equations, which is probably what would be most useful to readers of Scholarpedia. I don’t see articles planned on standard existence and uniqueness theory or on stability theory which are only mentioned in passing by this article. If such articles were written they would provide useful cross-references for this article. Articles related to these questions do appear in Wikpedia. Possibly they could be cross-referenced. Within the limitations chosen by the authors the article is generally well-written and should be within the grasp of a mathematically informed technical reader. Here are some suggestions for possible improvements:

    1. In the paragraph beginning “A boundary value…” replace “that can be used to solve it” by “above usually”. 2. Even though there is to be a separate article on stiff systems I think the authors need to define the term at least loosely at the point where it is introduced, that is in the paragraph on van der Pol equations. They need to say that most ODEs are non-stiff and then what distinguishes a stiff system. 3. In the paragraph beginning “Discrete variable…” the last sentence could be improved as follows: Methods with memory (Adams and BDF are examples) use some of these values; one-step methods (for example Runge-Kutta) evaluate f only at points in [t_n, t_{n+1}]. 4. In the sentence beginning “The backward Euler…” after “several ways” insert “with extensions”. 5. There is too much repetition of previous material about the backward Euler method in the Backward Differentiation Formulas section. 6. In the paragraph starting “How an implicit…” before the word “fixed”, for emphasis I would include “predetermined”. Again for emphasis, I would insert “effectively” before “done with an implicit”. 7. In the section on Runge-Kutta Methods the authors should consider writing the Heun method in standard Runge-Kutta form and writing down the expression for the local truncation error to emphasize the difference in form in comparison with Adams and BDF. 8. In the paragraph starting “On reaching…” possibly insert “it measures” after “i.e.”. There is a hanging period in this paragraph. 9. In the next paragraph insert “solving” before “non-stiff initial”. 10. In the paragraph staring “General-purpose…” insert “approach” after “This”. 11. The next paragraph could be improved with introductory sentences on the lines of “The truncation error in all the popular formulas involves high derivatives of the solution or differential equation. To estimate the truncation error these derivatives are estimated using methods which involve little or no additional computation.” I would drop the two sentences about BDF in favor of a sentence on the lines of “For BDF methods the high derivative of the solution is commonly estimated using a difference approximation based on past values of the solution.” 12. In the paragraph starting “The Adams methods” the implication is that the most efficient step size and order are actually computed rather than that they are estimated. Later in the paragraph I would put “values” after “more”. 13. There are more recent editions of both the quoted texts by Hairer and Wanner – see the Springer publications page.

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