Efficient numerical schemes for the approximation of expectations of functionals of the solution of a S.D.E., and applications

AbstractThis paper concerns the flow of fluid exiting a two-dimensional pipe and impacting an infinite wedge. Where the flow leaves the pipe there is a free surface between the fluid and a passive gas. The model is a generalisation of both plane bubbles and flow impacting a flat plate. In the absence of gravity and surface tension, an exact free streamline solution is derived. We also construct two numerical schemes to compute solutions with the inclusion of surface tension and gravity. The first method involves mapping the flow to the lower half-plane, where an integral equation concerning only boundary values is derived. This integral equation is solved numerically. The second method involves conformally mapping the flow domain onto a unit disc in the s-plane. The unknowns are then expressed as a power series in s. The series is truncated, and the coefficients are solved numerically. The boundary integral method has the additional advantage that it allows for solutions with waves in the far-field, as discussed later. Good agreement between the two numerical methods and the exact free streamline solution provides a check on the numerical schemes.

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PARALLEL COMPUTING OF NUMERICAL SCHEMES AND BIG DATA ANALYTIC FOR SOLVING REAL LIFE APPLICATIONS

Jurnal Teknologi ◽

10.11113/jt.v78.9552 ◽

2016 ◽

Vol 78 (8-2) ◽

Cited By ~ 2

Author(s):

Norma Alias ◽

Nadia Nofri Yeni Suhari ◽

Hafizah Farhah Saipan Saipol ◽

Abdullah Aysh Dahawi ◽

Masyitah Mohd Saidi ◽

...

Keyword(s):

Big Data ◽

Parallel Computing ◽

Parallel Algorithm ◽

Sparse Matrices ◽

Real Life ◽

Poor Performance ◽

Equation System ◽

Numerical Schemes ◽

Linear Equation System ◽

Data Analytic

This paper proposed the several real life applications for big data analytic using parallel computing software. Some parallel computing software under consideration are Parallel Virtual Machine, MATLAB Distributed Computing Server and Compute Unified Device Architecture to simulate the big data problems. The parallel computing is able to overcome the poor performance at the runtime, speedup and efficiency of programming in sequential computing. The mathematical models for the big data analytic are based on partial differential equations and obtained the large sparse matrices from discretization and development of the linear equation system. Iterative numerical schemes are used to solve the problems. Thus, the process of computational problems are summarized in parallel algorithm. Therefore, the parallel algorithm development is based on domain decomposition of problems and the architecture of difference parallel computing software. The parallel performance evaluations for distributed and shared memory architecture are investigated in terms of speedup, efficiency, effectiveness and temporal performance.

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GEOMETRIC INTEGRATION BY SOLUTION INTERPOLATION

International Journal of Modern Physics C ◽

10.1142/s0129183109013637 ◽

2009 ◽

Vol 20 (02) ◽

pp. 313-322

Author(s):

PILWON KIM

Keyword(s):

Differential Equation ◽

Heat Equation ◽

Exact Solutions ◽

Standard Method ◽

Interpolation Method ◽

Nonlinear Heat Equation ◽

Geometric Integration ◽

Numerical Schemes ◽

New Class ◽

Geometric Integrators

Numerical schemes that are implemented by interpolation of exact solutions to a differential equation naturally preserve geometric properties of the differential equation. The solution interpolation method can be used for development of a new class of geometric integrators, which generally show better performances than standard method both quantitatively and qualitatively. Several examples including a linear convection equation and a nonlinear heat equation are included.

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