Nonlinear Stability And Error Analysis Of General Linear Method For Two Classes Of Stiff Differential Equations

Stiff differential equations are widely used in many fields. Many scholars have devoted great energy research to them, and have achieved fruitful results.In the first chapter, we recall the origin of delay differential equations,and the development of the study of their numerical stability.Then,we turn our attention to singular perturbation problems (SPPs),which are a special class of stiff ordinary differential equations(ODEs).Some numerical methods and related convergence results are reviewed. Meanwhile,we briefly introduce the convergence theory of ordinary differential equations for numeical methods .In the second chapter, we discuss nonlinear stability of general linear methods for variable delay differential equations.Under the assumption that the delay satisfies Lipschitz condition with the minimum lipschitz constant L<1, we obtain some results on the stability of general linear methods with linear interpolation procedure.In the third chapter, we discuss general linear methods for solving a class of multiply stiff singular perturbation problems. On the basis of Gan's article, we relax the requirement of algebraic stability, under the weakly algebraic and diagonal stability conditions,and obtained the error estimates of general linear methods.In the fourth chapter, we discuss general linear method of solving two parameters singular perturbation problems(SPPs).On the basis of the work by Xiao on the convergence of Runge-Kutta methods and linear multistep methods for two parameters SPPs,combined with the convergence results on general linear methods for one parameter SPPs by Gan,we further obtain a global error estimate for general linear methods for two parameter SPPs.In the fifth chapter, some numerical tests are given to confirm the correctness of the conclusions.