A Problem-Dependent Formula of Damping for Stiff Problems

A matrix coefficient formula of damping is developed to solve stiff problems. This is a matrix coefficient formula that has L-stability, explicitness and 2nd-order accuracy. It possesses damping to suppress or eliminate the negligible stiff components of the problems under analysis. It corresponds to the 2-step backward differentiation formula in both performance and property. However, it can possess a noniterative and an iterative solution procedure. An iterative procedure of the formula will perform the same as the 2-step backward differentiation formula. In addition, its noniterative procedure can still perform well as a 2-step backward differentiation formula for general nonlinear problems. However, it may perform slightly worse than the 2-step backward differentiation formula for nonstiff highly nonlinear problems. Hence, the most significant improvement of the proposed formula in contrast to the 2-step backward differentiation formula is that it can have a noniterative process for each step and thus it has no convergent problem per step. As a result, it is an efficient formula to solve stiff problems with negligible fast components. In fact, it is numerically affirmed that it can save many computational efforts for such stiff problems of large systems of ODEs.