Finite difference method

Abstract: A CFD model is proposed for numerical simulations of extremely nonlinear free-surface flows such as wave impact phenomena and violent wave–body interactions. The constrained interpolation profile (CIP) method is adopted as the base scheme for the model. The wave–body interaction is treated as a multiphase problem, which has liquid (water), gas (air), and solid (wave-maker and floating body) phases. The flow is represented by one set of governing equations, which are solved numerically on a nonuniform, staggered Cartesian grid by a finite-difference method. The free surface as well as the body boundary are immersed in the computation domain and captured by different methods. In this article, the proposed numerical model is first described. Then to validate the accuracy and demonstrate the capability, several two-dimensional numerical simulations are presented, and compared with experiments and with computations by other numerical methods. The numerical results show that the present computation model is both robust and accurate for violent free-surface flows.

Abstract: A phase field model is used to numerically simulate the solidification of a pure material. We employ it to compute growth into an undercooled liquid for a one-dimensional spherically symmetric geometry and a planar two-dimensional rectangular region. The phase field model equation are solved using finite difference techniques on a uniform mesh. For the growth of a sphere, the solutions from the phase field equations for sufficiently small interface widths are in good agreement with a numerical solution to the classical sharp interface model obtained using a Green's function approach. In two dimensions, we simulate dendritic growth of nickel with four-fold anisotropy and investigate the effect of the level of anisotropy on the growth of a dendrite. The quantitative behavior of the phase field model is evaluated for varying interface thickness and spatial and temporal resolution. We find quantitatively that the results depend on the interface thickness and with the simple numerical scheme employed it is not practical to do computations with an interface that is sufficiently thin for the numerical solution to accurately represent a sharp interface model. However, even with a relatively thick interface the results from the phase field model show many of the features of dendritic growth and they are in surprisingly good quantitative agreement with the Ivantsov solution and microscopic solvability theory.

Abstract: The common techniques for solving two-point boundary value problems can be classified as either "shooting" or "finite difference" methods. Central to a shooting method is the ability to integrate the differential equations as an initial value problem with guesses for the unknown initial values. This ability is not required with a finite difference method, for the unknowns are considered to be the values of the true solution at a number of interior mesh points. Each method has its advantages and disadvantages. One serious shortcoming of shooting becomes apparent when, as happens altogether too often, the differential equations are so unstable that they "blow up" before the initial value problem can be completely integrated. This can occur even in the face of extremely accurate guesses for the initial values. Hence, shooting seems to offer no hope for some problems. A finite difference method does have a chance for it tends to keep a firm hold on the entire solution at once. The purpose of this note is to point out a compromising procedure which endows shooting-type methods with this particular advantage of finite difference methods. For such problems, then, all hope need not be abandoned for shooting methods. This is desirable because shooting methods are generally faster than finite difference methods.The organization is as follows: I. The two-point boundary value problem is stated in quite general form. II. A particular shooting method is described which is designed to solve the problem in this form. III. The two-point boundary value problem is then restated in such a way that: (a) the restatement still falls within the general form, and (b) the shooting method now has a better chance of success when the equations are unstable.