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The original 3D-ADI method implies dividing each time step into three minor, consecutive time sub-steps equal in size. In the modication by Douglas & Gunn the three time sub-steps are not equal in size and should hence, to some extend, be interpreted as overlapping - this is illustrated in g. 2.3. ADI method application for 2D problems Real-time Depth-Of-Field simulation —Using diffusion equation to blur the image Now need to solve tridiagonal systems in 2D domain —Different setup, different methods for GPU In numerical linear algebra, the Alternating Direction Implicit (ADI) method is an iterative method used to solve Sylvester matrix equations. It is a popular method for solving the large matrix equations that arise in systems theory and control, and can be formulated to construct solutions in a memory-efficient, factored form. neous convection–diffusion equation [16] and a three-dimensional (3D) homogeneous heat equation [17]. However, grand difﬁculties are encountered when the IIM–ADI method [14,16,17]is generalized in [15] to solve a 2D heat equation with nonhomogeneous media, i.e., α being a piecewise constant.

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6.1 - ADI Method, a Fast Implicit Method for 3D USS HT The Alternating Direction Implicit (ADI) Method of solving PDQ’s is based on the Crank-Nicolson Method of solving one-dimensional problems. ADI method – iterations Use global iterations for the whole system of equations Some equations are not linear: —Use local iterations to approximate the non-linear term previous time step Solve X-dir equations Solve Y-dir equations Solve Z-dir equations Updating all variables next time step global iterations Project - Solving the Heat equation in 2D. 303 Linear Partial Diﬀerential Equations Matthew J. (See example codes). m to solve the 2D heat equation using the explicit approach. Example of ADI method foe 2D heat equation this is a matlab code of the method of visual cryptography based in the. ADI&Scheme& (19.4) For the second step from equation (19.1) is approximated by Rearranging gives (19.5) which is implicit. When writing for a 2-dimensional grid, the equation results in a tri-diagonal system. ADI method – iterations Use global iterations for the whole system of equations Some equations are not linear: —Use local iterations to approximate the non-linear term previous time step Solve X-dir equations Solve Y-dir equations Solve Z-dir equations Updating all variables next time step global iterations • Alternating Direction Implicit (ADI)! • Approximate Factorization of Crank-Nicolson! Splitting! Outline! Solution Methods for Parabolic Equations! Computational Fluid Dynamics! Numerical Methods for! One-Dimensional Heat Equations! Computational Fluid Dynamics! taxb x f t f ><< ∂ ∂ = ∂ ∂;0, 2 2 α which is a parabolic equation ...

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