For fourth-order geometric evolution equations for planar curves with the dissipation of the bending energy,including the Willmore and the Helfrich flows,we consider a numerical approach.In this study,we construct a s...For fourth-order geometric evolution equations for planar curves with the dissipation of the bending energy,including the Willmore and the Helfrich flows,we consider a numerical approach.In this study,we construct a structure-preserving method based on a discrete variational derivative method.Furthermore,to prevent the vertex concentration that may lead to numerical instability,we discretely introduce Deckelnick’s tangential velocity.Here,a modification term is introduced in the process of adding tangential velocity.This modified term enables the method to reproduce the equations’properties while preventing vertex concentration.Numerical experiments demonstrate that the proposed approach captures the equations’properties with high accuracy and avoids the concentration of vertices.展开更多
In this paper two classes of equivalence transform methods for solving ordinary differential equations are proposed. One class of method is the equivalence integral transform method for special differential algebraic ...In this paper two classes of equivalence transform methods for solving ordinary differential equations are proposed. One class of method is the equivalence integral transform method for special differential algebraic problems. The advantage of this class of method is such that the amount of work calculating one integration with parameters becomes that of two interpolations, when the system of nonlinear equations is solved on the right hand side function. The other class of method is the equivalence substitution method for avoiding calculating derivative on the right hand side function. In order to avoid calculation derivatives, two equivalence substitution methods are proposed here. The application instances of some special effect of the equivalence substitution methods are given.展开更多
基金This work was supported by JSPS KAKENHI Grant Nos.19K14590,21K18301,Japan.
文摘For fourth-order geometric evolution equations for planar curves with the dissipation of the bending energy,including the Willmore and the Helfrich flows,we consider a numerical approach.In this study,we construct a structure-preserving method based on a discrete variational derivative method.Furthermore,to prevent the vertex concentration that may lead to numerical instability,we discretely introduce Deckelnick’s tangential velocity.Here,a modification term is introduced in the process of adding tangential velocity.This modified term enables the method to reproduce the equations’properties while preventing vertex concentration.Numerical experiments demonstrate that the proposed approach captures the equations’properties with high accuracy and avoids the concentration of vertices.
基金The project was supported by the National Natural Science Faundation of China
文摘In this paper two classes of equivalence transform methods for solving ordinary differential equations are proposed. One class of method is the equivalence integral transform method for special differential algebraic problems. The advantage of this class of method is such that the amount of work calculating one integration with parameters becomes that of two interpolations, when the system of nonlinear equations is solved on the right hand side function. The other class of method is the equivalence substitution method for avoiding calculating derivative on the right hand side function. In order to avoid calculation derivatives, two equivalence substitution methods are proposed here. The application instances of some special effect of the equivalence substitution methods are given.
