Adaptive Moving Grid Methods For Singularly Perturbed Volterra Integro-differential Equations

Singularly perturbed Volterra integro-differential equations are widely used in science and engineering.Because most singularly perturbed Volterra integro-differential equations are difficult or even impossible to obtain their exact solutions,their numerical methods have attracted the interest of many scholars.Adaptive moving grid method has been widely used to solve some singularly perturbed differential equations,and there are relatively perfect numerical methods.However,for singularly perturbed Volterra integro-differential equations,the application of adaptive grid method are relatively few.Based on this,this dissertation mainly studies the adaptive moving grid method for singularly perturbed Volterra integro-differential equation and systems of singularly perturbed Volterra integro-differential equations,aiming at enriching the practical application of this kind of problems.The details are as follows:Chapter ? is the introduction,which introduces the research background and progress of singularly perturbed Volterra integro-differential problems,and briefly introduces the main work of this dissertation.In Chapter ?,an adaptive moving mesh method for a nonlinear singularly perturbed Volterra integro-differential equation is considered.Firstly,we construct the upwind finite difference scheme for this kind of problems,and derive the corresponding a priori and a posteriori error estimates.Then,an adaptive moving mesh generation algorithm is designed basing on the a posteriori error estimates.The numerical results verify the first-order uniform convergence of the numerical method.In Chapter ?,a relatively simple finite difference scheme is constructed for systems of singular perturbed Volterra integro-differential equations.Furthermore,the a posterior error estimation and the corresponding mesh generation algorithm are given.Finally,a linear and nonlinear test problem are given to verify the theoretical results of this chapter.The numerical results show that the adaptive mobile grid method is very effective.In Chapter ?,a second-order adaptive moving grid method for singularly perturbed Volterra integro-differential equation is considered.Firstly,we construct a discrete scheme for this kind of problem,and derive the truncation error estimate of the discrete scheme.Based on this,we design a suitable monitor function and derive a posteriori error estimate,and then give a moving grid generation algorithm.A numerical example is given to verify the theory and algorithm proposed in this chapter.Finally,numerical results show the second order uniform convergence of the numerical method.