# lect_16 - Numerical Marine Hydrodynamics Numerical...

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Numerical Marine Hydrodynamics Numerical Differentiation – Newton Interpolation – Finite Differences Ordinary Differential Equations – Initial Value Problems • Euler’s Method • Taylor Series Methods – Error analysis • Predictor-Corrector Methods • Runge-Kutta Methods • Stiff Differential Equations • Multistep Methods • Error Analysis and Error Modifiers – Systems of differential equations – Boundary Value Problems • Shooting method • Direct Finite Difference methods Numerical Marine Hydrodynamics Lecture 16 2.29

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Initial Value Problems Error Analysis Integrate Trapezoidal Rule Heun’s Corrector Trapezoidal Rule Error Initial Value Problem Heun’s non-Self-starter Predictor Mid-point Integration Heun’s non Self-Starter Predictor Mid-point Integration Error Numerical Marine Hydrodynamics Lecture 16 2.29
Smooth y’’’ Initial Value Problems Error Modifiers Predictor Corrector Same Order Subtract Corrector Error Corrector Modifier Predictor Error Predictor Modifier Replace by Numerical Marine Hydrodynamics Lecture 16 2.29

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Initial Value Problems Higher Order Differential Equations Differential Equation Initial Conditions Matrix form Convert to 1 st Order System Solved using e.g. Runge-Kutta (ode45) Numerical Marine Hydrodynamics Lecture 16 2.29
Sphere Motion in Fluid Flow MATLAB Solutions V x R M dx u = dt function [f] = dudt(t,u) dudt.m % u(1) = u % u(2) = x % f(2) = dx/dt = u % f(1) = du/dt=rho*Cd*pi*r/(2m)*(v^2-2uv+u^2) [rho,Cd,m,r,v] = sph_param(); fac=rho*Cd*pi*r^2/(2*m); f(1)=fac*(v^2-2*u(1)+u(1)^2); f(2)=u(1); f=f'; %step size h=1.0; sph_drag_2.m % Euler's method, forward finite difference t=[0:h:10];

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## This note was uploaded on 02/27/2012 for the course MECHANICAL 2.29 taught by Professor Henrikschmidt during the Spring '07 term at MIT.

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lect_16 - Numerical Marine Hydrodynamics Numerical...

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