EFFICIENT STABILITY-PRESERVING NUMERICAL METHODS FOR NONLINEAR COERCIVE PROBLEMS IN VECTOR SPACE.

Strong stability (or monotonicity)-preserving time discretization schemes preserve the stability properties of the exact solution and have proved very useful in scientific and engineering computation, especially in solving hyperbolic partial differential equations. The main aim of this work is to further extend this to exponential stability-preserving numerical methods for a general coercive system whose solution is exponentially growing or decaying and the rate of growth or decay can be quantified by a (ω,τ*) function in general vector space with a convex functional. Under the same stepsize condition as for strong stability, sharper exponential stability results are derived for explicit and diagonally implicit Runge-Kutta methods and variable coefficients linear multistep methods for nonlinear problems. The new developments in this paper also include their applications to various linear and nonlinear evolution problems. [ABSTRACT FROM AUTHOR]