4
$\begingroup$

I have an ODE for a scalar function $u=u(t)$ of the form: $$ \frac{du}{dt}=L(u). $$ Here the function $L=L(u)$ satisfies: $$ L(0)=0, \quad L'(u)\le0. $$ Then it is easy to see that the solution $u=u(t)$ has the following property:

(i) If the initial value $u(0)\ge0$, then $u(t)\ge0$ for any $t>0$;

(ii) If the initial value $-M\le u(0)\le M$ with $M>0$, then $-M\le u(t)\le M$.

Question: I need to find a numerical method to solve this ODE and under arbitrary step size maintain the two properties or only one of two. I only know that the Euler backward method has these two properties. Does anyone of you know high order (e.g. third-order) methods to solve this problem?

Any links to literature or for further reading would be greatly appreciated.

$\endgroup$
  • $\begingroup$ @DavidKetcheson The solution of an ODE has the property of monotonicity-preserving, i.e., if $u(t)$ and $v(t)$ are solutions to the same ODE and $u(0)\le v(0)$, then $u(t)\le v(t)$ for any $t>0$. Since $u\equiv0$ is a solution to this ODE, property (i) holds for both positivity-preserving and negativity-preserving with the aid of monotonicity-preserving property. $\endgroup$ – Michael Jun 16 '16 at 18:34
6
$\begingroup$
$\endgroup$
  • 2
    $\begingroup$ There are also a more recent review and a book (disclaimer: I'm an author on both). The review is here: brown.edu/research/projects/scientific-computing/sites/…. $\endgroup$ – David Ketcheson Jun 16 '16 at 17:03
  • 1
    $\begingroup$ The result regarding backward Euler goes back to a 1979 paper of Bolley and Crouzeix (in French). $\endgroup$ – David Ketcheson Jun 16 '16 at 17:04
  • $\begingroup$ Thanks for your wonderful answers! I just reviewed these literature briefly. In summary, it seems that there is NO ode solver which is greater than first-order and has the positivity-preserving and/or monotonicity-preserving property under arbitrary step size. $\endgroup$ – Michael Jun 16 '16 at 18:38
  • $\begingroup$ @Michael See also this paper (reference [79] in the review David Ketcheson linked to): math.leidenuniv.nl/~spijker/PUBLICATIONS-GENERAL/… $\endgroup$ – Kirill Jun 16 '16 at 18:45
  • $\begingroup$ @Kirill Thanks for the reference. I went through this paper and found in Section 1.4 Remark in the first paragraph: the authors explained that, the contractivity property concerns arbitrary pairs satisfying (1.2) but not for some pairs of specific form. I think the conclusion in that paper does not completely rule out the existence of solver for the ODE in my problem. $\endgroup$ – Michael Jun 16 '16 at 18:52

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.