Stiff dynamical systems numerical simulation

Lu et al. (2009) identified QSS-species and pre-equilibrium reactions on the fly, based on the investigation of system timescales. This information was used to convert the original system of differential equations to a less stiff system of differential algebraic equations. This dynamic stiffness removal method for accelerating simirlations was successfully applied for predictions using an n-heptane oxidatirm mechanism in ID and 2D turbulent direct numerical simulations. 

Usually, in numerically simulated dynamic systems, such as multistory buildings under earthquake excitations, structural impact is considered using force-based methods, also known as penalty methods. These methods allow relatively small interpenetration between the colliding strucmres, which can be justified by the local deformability at the point of impact. The interpenetration depth is used together with an impact stiffness coefficient, which represents an impact spring, to calculate the impact forces that act on the colliding structures and push them apart. 

However, often the real problem is not with the numerical algorithm but with the engineer developing the equation set. If one is interested in the slower dynamic parts of the problem, a quasi-steady-state assumption should be made for the fast parts of the problem. On the other hand, if one is interested in the fast parts of the problem, the value of the slower parts essentially remains constant over these very short time periods. Therefore, stiff systems of equations should not arise in most properly formulated simulations that use order-of-magnitude scaling in model formation. 

Molecular Dynamics (MD) is the most fundamental approach to soft-matter simulations. Here the solute particles are immersed in a bath of solvent molecules and Newton s equations of motion are solved numerically. In this case, it is impossible to make the solvent structureless - a structureless solvent would be an ideal gas of point particles, which never reaches thermal equilibrium. Furthermore, the model interaction potentials are stiff and considerable simulation time is spent following the motion of the solvent particles in their local cages. These disadvantages are so severe that nowadays MD is rarely applied to soft-matter systems of the type we are discussing in this article. 

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