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
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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
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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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lect_16 - Numerical Marine Hydrodynamics Numerical...

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