Positive Numerical Integration Methods for Chemical Kinetic Systems

Abstract

Chemical kinetics conserves mass and renders nonnegative solutions; a good numerical simulation would ideally produce mass-balanced, positive concentration vectors. Many time-stepping methods are mass conservative; however, unconditional positivity restricts the order of a traditional method to one. The projection method presented in this paper ensures mass conservation and positivity. First, a numerical approximation is computed with one step of a mass-preserving traditional scheme. If there are negative components, the nearest vector in the reaction simplex is found by solving a quadratic optimization problem; this vector is shown to better approximate the true solution. A simpler version involves just one projection step and stabilizes the reaction simplex. This technique works best when the underlying time-stepping scheme favors positivity. Projected methods are more accurate than clipping and allow larger time steps for kinetic systems which are unstable outside the positive quadrant.