On the positivity of low order explicit Runge-Kutta schemes applied in splitting methods

Splitting methods are a frequently used approach for the solution of large stiff initial value problems of ordinary differential equations with an additively split right-hand side function. Such systems arise, for instance, as method of lines discretizations of evolutionary partial differential equations in many applications. We consider the choice of explicit Runge-Kutta (RK) schemes in implicit-explicit splitting methods. Our main objective is the preservation of positivity in the numerical solution of linear and nonlinear positive problems while maintaining a sufficient degree of accuracy and computational efficiency. A 3-stage second order explicit RK method is proposed which has optimized positivity properties. This method compares well with standard s-stage explicit RK schemes of order s, s = 2; 3. It has advantages in the low accuracy range and this range is interesting for an application in splitting methods. Numerical results are presented.