Finite difference method

Abstract: This paper examines different approaches for driving mesh adaptation and provides theoretical developments for understanding the relationship between discretization error, the numerical scheme, and the mesh. Discrete and continuous equations governing the transport of discretization error are developed and it is shown that the truncation error acts as the local source for these equations. Examination of the truncation error in generalized coordinates provides insight into the role of mesh quality (mesh stretching for the 1D case) in the discretization error. Numerical results are presented for 1D steady-state Burgers equation at Reynolds numbers of 32 and 128. Four different approaches for driving mesh adaption are implemented for this case: solution gradients, solution curvature, discretization error, and truncation error. The truncation-error based adaption is shown to provide superior results for both cases. Finally, two approaches for estimating the truncation error are also discussed which would allow truncation error-based adaption to be implemented for complex numerical methods.

Abstract: Conservative linear equations arise in many areas of application, including continuum mechanics or high-frequency geometrical optics approximations. This kind of equations admits most of the time solutions which are only bounded measures in the space variable known as duality solutions. In this paper, we study the convergence of a class of finite-differences numerical schemes and introduce an appropriate concept of consistency with the continuous problem. Some basic examples including computational results are also supplied.

Abstract: An alternative artificial compressibility (AC) scheme is proposed to allow the explicit simulation of the incompressible Navier-Stokes (INS) equations. Traditional AC schemes rely on an artificial equation of state that gives the pressure as a function of the density, which is known to enforce isentropic behavior. This behavior is nonideal, especially in viscously dominated flows. An alternative, the entropically damped artificial compressibility (EDAC) method, is proposed that employs a thermodynamic constraint to damp the pressure oscillations inherent to AC methods. The EDAC method converges to the INS in the low-Mach limit, and is consistent in both the low- and high-Reynolds-number limits, unlike standard AC schemes. The proposed EDAC method is discretized using a simple finite-difference scheme and is compared with traditional AC schemes as well as the lattice-Boltzmann method for steady lid-driven cavity flow and a transient traveling-wave problem. The EDAC method is shown to be beneficial in damping pressure and velocity-divergence oscillations when performing transient simulations. The EDAC method follows a similar derivation to the kinetically reduced local Navier-Stokes (KRLNS) method [Borok et al., Phys. Rev. E 76, 066704 (2007)]; however, the EDAC method does not rely on the grand potential as the thermodynamic variable, but instead uses the more common pressure-velocity system. Additionally, a term neglected in the KRLNS is identified that is important for accurately approximating the INS equations.

Abstract: A partitioned approach by the coupling finite difference method (FDM) and the finite element method (FEM) is developed for simulating the interaction between free surface flow and a thin elastic plate. The FDM, in which the constraint interpolation profile method is applied, is used for solving the flow field in a regular fixed Cartesian grid, and the tangent of the hyperbola for interface capturing with the slope weighting scheme is used for capturing free surface. The FEM is used for solving structural deformation of the thin plate. A conservative momentum-exchange method, based on the immersed boundary method, is adopted to couple the FDM and the FEM. Background grid resolution of the thin plate in a regular fixed Cartesian grid is important to the computational accuracy by using this method. A virtual structure method is proposed to improve the background grid resolution of the thin plate. Both of the flow solver and the structural solver are carefully tested and extensive validations of the coupled FDM–FEM method are carried out on a benchmark experiment, a rolling tank sloshing with a thin elastic plate.