Finite difference

About: Finite difference is a research topic. Over the lifetime, 19693 publications have been published within this topic receiving 408603 citations.

Realistic animation of liquids

Abstract: We present a comprehensive methodology for realistically animating liquid phenomena Our approach unifies existing computer graphics techniques for simulating fluids and extends them by incorporating more complex behavior It is based on the Navier–Stokes equations which couple momentum and mass conservation to completely describe fluid motion Our starting point is an environment containing an arbitrary distribution of fluid, and submerged or semisubmerged obstacles Velocity and pressure are defined everywhere within this environment and updated using a set of finite difference expressions The resulting vector and scalar fields are used to drive a height field equation representing the liquid surface The nature of the coupling between obstacles in the environment and free variables allows for the simulation of a wide range of effects that were not possible with previous computer graphics fluid models Wave effects such as reflection, refraction, and diffraction, as well as rotational effects such as eddies, vorticity, and splashing are a natural consequence of solving the system In addition, the Lagrange equations of motion are used to place buoyant dynamic objects into a scene and track the position of spray and foam during the animation process Typical disadvantages to dynamic simulations such as poor scalability and lack of control are addressed by assuming that stationary obstacles align with grid cells during the finite difference discretization, and by appending terms to the Navier–Stokes equations to include forcing functions Free surfaces in our system are represented as either a collection of massless particles in 2D, or a height field which is suitable for many of the water rendering algorithms presented by researchers in recent years

The heterogeneous multiscale method

Abstract: The heterogeneous multiscale method (HMM), a general framework for designing multiscale algorithms, is reviewed. Emphasis is given to the error analysis that comes naturally with the framework. Examples of finite element and finite difference HMM are presented. Applications to dynamical systems and stochastic simulation algorithms with multiple time scales, spall fracture and heat conduction in microprocessors are discussed.

A Boundary Condition Capturing Method for Multiphase Incompressible Flow

Abstract: In l6r, the Ghost Fluid Method (GFM) was developed to capture the boundary conditions at a contact discontinuity in the inviscid compressible Euler equations. In l11r, related techniques were used to develop a boundary condition capturing approach for the variable coefficient Poisson equation on domains with an embedded interface. In this paper, these new numerical techniques are extended to treat multiphase incompressible flow including the effects of viscosity, surface tension and gravity. While the most notable finite difference techniques for multiphase incompressible flow involve numerical smearing of the equations near the interface, see, e.g., l19, 17, 1r, this new approach treats the interface in a sharp fashion.

Calculation of plane steady transonic flows

Abstract: Transonic small disturbance theory is used to solve for the flow past thin airfoils including cases with imbedded shock waves. The small disturbance equations and similarity rules are presented, and a boundary value problem is formulated for the case of a subsonic freestream Mach number. The governing transonic potential equation is a mixed (elliptic-hyperbolic) differential equation which is solved numerically using a newly developed mixed finite difference system. Separate difference formulas are used in the elliptic and hyperbolic regions to account properly for the local domain of dependence of the differential equation. An analytical solution derived for the far field is used as a boundary condition for the numerical solution. The difference equations are solved with a line relaxation algorithm. Shock waves, if any, and supersonic zones appear naturally during the iterative process. Results are presented for nonlifting circular arc airfoils and a shock free Nieuwland airfoil. Agreement with experiment for the circular arc airfoils, and exact theory for the Nieuwland airfoil is excellent.