Solve an ODE with positivity-preserving property unconditionally

Question

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.

Answer 1

• The two properties are usually called positivity-preserving and monotonicity-preserving (makes it easier to find this question).

• Looking at http://www.ams.org/journals/mcom/2006-75-254/S0025-5718-05-01794-1/S0025-5718-05-01794-1.pdf and http://homepages.cwi.nl/~willem/DOCART/SIAM_HRS.pdf it seems that implicit Euler is the exception among implicit linear multistep methods in that only implicit Euler allows for timesteps to be arbitrary while preserving monotonicity (section 5.3). They say the restrictions on implicit linear multistep methods' timesteps are not much better than on explicit multistep methods.

• There is a discussion of monotonicity-preserving Runge-Kutta methods in Hairer-Wanner Vol.II, Section IV.11: implicit Euler is the only one among them having a threshold factor of $\infty$.

• See also http://www.cscamm.umd.edu/tadmor/pub/linear-stability/Gottlieb-Shu-Tadmor.SIREV-01.pdf, which discusses high-order strong stability-preserving methods and gives an overview of available methods (not unconditionally SSP, though).