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 general coercive system whose solution is exponentially growing or decaying and the rate of growth or decay can be quantified by a $(\omega,\tau^*)$ 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.

王晚生，上海师范大学教授，博导，数理学院副院长，数学科学研究所所长。主要从事微分方程数值解法及应用方面的教学研究工作，在金融期权模型理论分析和快速算法、微分方程保稳定性算法和自适应算法、数据同化和深度学习算法等方面取得了一系列成果。曾入选湖南省新世纪“121人才工程”、湖南省普通高校学科带头人等人才计划，系AAMM编委、中国仿真学会理事、中国工业与应用数学学会金融科技与算法专委会常务委员、中国数学会计算数学分会理事等。