First Half Topics
Explain 0D, 1D, 2D, and 3D models
-0D models are only a function of time (f(t)) -1D models are a function of time and space (f(t,x)) -2D models are a function of time and space (f(x,y,t)) -3D models are a function of time and space (f(x,y,z,t))
What can be modeled with a 1D model?
A river in which we only care about a factor in one direction and spatial features can be neglected
What is the first rule consistency?
A scheme must be selected such that it approximates in some way the ODE in small spatial time steps
What does accuracy depend on?
Accuracy depends on tolerance and we set the tolerance. It is independent on pace and robustness
What is a way to fix the pitfall of the bisection method?
By using the false position method which joins f(xl) and f(xu) by a straight line as a means of determining the next guess where the x axis is crossed.
What are the three golden rules of numerical simulation?
Consistency, Convergence, stability
What are pitfalls of the newton raps hon method?
Convergence depends on the nature of the function and on the accuracy of the starting initial guess
Explain the process of the fixed point method.
First solve for an x (gx)
Are the threee methods explicit or implicit?
Forward is explicit while backward and central are implicit
When is it said that the mesh selected is consistent?
If a scheme is selected such that the equation tends to the differential equation in the process of refining the mesh
When is a numerical solution stable?
If the amplification of the error remains bounded
What are pitfalls of the false position method?
It is largely influenced by the shape of the function, and since one of the bracketing roots tends to stay fixed it can lead to poor convergence
What can be modeled with a 3D model?
Lake Tahoe where depth must be considered
What is the meaning of simulation? What are some examples of simulation?
Simulation is the process of studying phenomena under a controlled conditions. Experiments, analog solutions, numerical methods
What does robust mean?
That a solution can be obtained regardless of the nature of the function
What does a fourth order accurate term mean?
That the error will be reduced by a factor of h^4
What is important about stability?
That the solution "makes sense" and is free from "wiggles"
What can be modeled with a 2D model?
The SF bay where the behavior of two spatial factors must be considered such as the flow in two directions
What are two bracketing methods?
The bisection and false position method
What happens as error accumulates?
The solution turns unstable
What are the traits of bracketing methods?
They are slow but robust
How can numerical stability be studied?
Through the analysis of the behavior of the error between the approximation and the exact solution for a large number of time steps.
What is a bracketing method consist of?
Two initial guesses that must bracket the root.
What is a pitfall of the bisection method?
When dividing the inerval of xl and xu into equal halves, no account is taken of the magnitude of f(xl) and f(xu)
What is convergence?
When the numerical results are close enough to the exact value.
What do you achieve by Ising the central method?
You get rid of error caused by curvature
What is the equation for the forward difference method? What order of accuracy is it?
f'i= (fi+1 - fi)/h + HOT -First order accurate
What is the equation for the backward difference method? What order of accuracy is it?
fi = (fi - fi-1)/h + HOT - First order accurate
What is the equation for the central method? What order of accuracy is it?
fi = (fi+1 - fi-1)/2h -Second order accurate