Hirofumi Notsu

Development and L2-Analysis of a Single-Step Characteristics Finite Difference Scheme of Second Order in Time for Convection-Diffusion Problems:

A new finite difference scheme based on the method of characteristics is presented for convection-diffusion problems. The scheme is of single-step and second order in time, and the matrix of the derived system of linear equations is symmetric. Since it is a finite difference scheme, we can get rid of numerical integration which may cause some instability in the characteristics finite element method. An optimal error estimate is proved in the framework of the discrete L2-theory. Numerical results are shown to recognize the convergence order and advantages of the scheme.

Error Estimates of a Stabilized Lagrange–Galerkin Scheme of Second-Order in Time for the Navier–Stokes Equations

Error estimates with optimal convergence orders are proved for a stabilized Lagrange–Galerkin scheme of second-order in time for the Navier–Stokes equations. The scheme is a combination of Lagrange–Galerkin method and Brezzi–Pitkaranta’s stabilization method. It maintains the advantages of both methods; (i) It is robust for convection-dominated problems and the system of linear equations to be solved is symmetric. (ii) Since the P1 finite element is employed for both velocity and pressure, the number of degrees of freedom is much smaller than that of other typical elements for the equations, e.g., P2/P1. Therefore, the scheme is efficient especially for three-dimensional problems. The second-order accuracy in time is realized by Adams-Bashforth’s (two-step) method for the discretization of the material derivative along the trajectory of fluid particles. The theoretical convergence orders are recognized by two- and three-dimensional numerical results.

A self-organized mesh generator using pattern formation in a reaction–diffusion system

Abstract A new type of mesh generator is developed by using a self-organized pattern in a reaction–diffusion system. The system is the Gray–Scott model, which creates a spot pattern in a specific parameter region. The spots correspond to nodes of a mesh. The mesh generator has several advantages: the algorithm is simple and processes to improve the mesh, such as smoothing, (locally) addition, and removal of nodes, are automatically performed by the system.

The Gradient Flow Structure of an Extended Maxwell Viscoelastic Model and a Structure-Preserving Finite Element Scheme

An extended Maxwell viscoelastic model with a relaxation parameter is studied from mathematical and numerical points of view. It is shown that the model has a gradient flow property with respect to a viscoelastic energy. Based on the gradient flow structure, a structure-preserving time-discrete model is proposed and existence of a unique solution is proved. Moreover, a structure-preserving P1/P0 finite element scheme is presented and its stability in the sense of energy is shown by using its discrete gradient flow structure. As typical viscoelastic phenomena, two-dimensional numerical examples by the proposed scheme for a creep deformation and a stress relaxation are shown and the effects of the relaxation parameter are investigated.

Stabilized Lagrange–Galerkin Schemes of First- and Second-Order in Time for the Navier–Stokes Equations

Two stabilized Lagrange–Galerkin schemes for the Navier–Stokes equations are reviewed. The schemes are based on a combination of the Lagrange–Galerkin method and Brezzi–Pitkaranta’s stabilization method. They maintain the advantages of both methods: (i) They are robust for convection-dominated problems and the systems of linear equations to be solved are symmetric; and (ii) Since the P1 finite element is employed for both velocity and pressure, the numbers of degrees of freedom are much smaller than that of other typical elements for the equations, e.g., P2/P1. Therefore, the schemes are efficient especially for three-dimensional problems. The one of the schemes is of first-order in time by Euler’s method and the other is of second-order by Adams–Bashforth’s method. In the second-order scheme an additional initial velocity is required. A convergence analysis is done for the choice of the velocity obtained by the first-order scheme, whose theoretical result is also recognized numerically.