This article describes a number of velocity-based moving mesh numerical methods for multidimensional nonlinear time-dependent partial differential equations(PDEs).It consists of a short historical review followed by a...This article describes a number of velocity-based moving mesh numerical methods for multidimensional nonlinear time-dependent partial differential equations(PDEs).It consists of a short historical review followed by a detailed description of a recently developed multidimensional moving mesh finite element method based on conservation.Finite element algorithms are derived for both mass-conserving and non mass-conserving problems,and results shown for a number of multidimensional nonlinear test problems,including the second order porous medium equation and the fourth order thin film equation as well as a two-phase problem.Further applications and extensions are referenced.展开更多
A distributed Lagrangian moving-mesh finite element method is applied to problems involving changes of phase.The algorithm uses a distributed conservation principle to determine nodal mesh velocities,which are then us...A distributed Lagrangian moving-mesh finite element method is applied to problems involving changes of phase.The algorithm uses a distributed conservation principle to determine nodal mesh velocities,which are then used to move the nodes.The nodal values are obtained from an ALE(Arbitrary Lagrangian-Eulerian)equation,which represents a generalization of the original algorithm presented in Applied Numerical Mathematics,54:450–469(2005).Having described the details of the generalized algorithm it is validated on two test cases from the original paper and is then applied to one-phase and,for the first time,two-phase Stefan problems in one and two space dimensions,paying particular attention to the implementation of the interface boundary conditions.Results are presented to demonstrate the accuracy and the effectiveness of the method,including comparisons against analytical solutions where available.展开更多
文摘This article describes a number of velocity-based moving mesh numerical methods for multidimensional nonlinear time-dependent partial differential equations(PDEs).It consists of a short historical review followed by a detailed description of a recently developed multidimensional moving mesh finite element method based on conservation.Finite element algorithms are derived for both mass-conserving and non mass-conserving problems,and results shown for a number of multidimensional nonlinear test problems,including the second order porous medium equation and the fourth order thin film equation as well as a two-phase problem.Further applications and extensions are referenced.
基金This work was undertaken with the support of EPSRC Grant EP/D058791/1.R.Mahmood wishes to thank his employer PINSTECH for granting him study leave to carry out research work at Leeds.
文摘A distributed Lagrangian moving-mesh finite element method is applied to problems involving changes of phase.The algorithm uses a distributed conservation principle to determine nodal mesh velocities,which are then used to move the nodes.The nodal values are obtained from an ALE(Arbitrary Lagrangian-Eulerian)equation,which represents a generalization of the original algorithm presented in Applied Numerical Mathematics,54:450–469(2005).Having described the details of the generalized algorithm it is validated on two test cases from the original paper and is then applied to one-phase and,for the first time,two-phase Stefan problems in one and two space dimensions,paying particular attention to the implementation of the interface boundary conditions.Results are presented to demonstrate the accuracy and the effectiveness of the method,including comparisons against analytical solutions where available.
