7) Numerical Solutions of PDEs Flashcards
When is a function said to be O(h^n)
What is the Foward Difference Method
Describe the proof of how the forward difference works
When is a finite-difference method said
be nth order
If the remainder term is O(h^n)
What is the Centred Differencing Method
Describe the proof of the Centred Differencing Method
What is the advantage of using centred differencing
It is higher order, but still only requires the same number of evaluations of u
How do you construct a finite-difference approximation for the first derivative of a function using Taylor’s theorem
Why are the coefficient conditions a+b+c+d=0,
a−c−2d=1, and a+c+4d=0 necessary in finite-difference formulas for first and second-order accuracy
First-Order Accuracy:
* Sum Zero (a+b+c+d=0): Ensures cancellation of constant terms, focusing the formula on the derivative approximation.
* Linear Coefficient (a−c−2d=1): Scales the first derivative term to 1, aligning the approximation with u′(x)
Second-Order Accuracy:
* Quadratic Coefficient Zero (a+c+4d=0): Removes the second derivative term from the error, reducing it to O(h^2 ) and increasing accuracy.
How is the second derivative of a function approximated using the central finite-difference method, and what is its accuracy
What is the Explicit Euler Method
Describe the proof of the Explicit Euler Method
What is the Implicit Euler Method
What is the Theta Method
How does the theta method generalise explicit and implicit Euler methods
Θ = 1 => Explicit Euler - (Error 0(∆t)
Θ = 0 => Implicit Euler - (Error 0(∆t)
Θ = 1/2 => Trapezoidal method - (Error 0(∆t^2)
What is the Trapezoidal Method
How is the Implicit Euler method applied to vector-valued ODEs
What is a Boundary Value Problem (BVP)
A PDE with two boundary conditions (often either the values at the boundaries or the derivatives at the boundaries)
How can finite-difference methods be used to approximate the solution of a boundary value problem (BVP)
How is the vector b structured when adding forcing terms to finite-difference schemes for BVPs
What is Global Error
The difference between the finite-difference approximation and the true solution, given at the gridpoints xj .
That is, ej = u(xj ) − uj , j = 0, 1, . . . , N
When is a finite-difference method said to be convergent
If it is both stable and
consistent or
What is the Truncation Error of a finite-difference relation
The local truncation error Tj at the grid point xj is the
remainder when uj is replaced by u(xj ) in the finite-difference relation for that point
When is a finite-difference method said to be consistent
A finite-difference method is said to be k-th order consistent for k > 0 if the local truncation error satisfies:
Tj = O(h^k)
What are the consistency orders of forward, backward and central finite-difference approximations
The forward and backward finite-difference approximations of the first spatial derivative are 1st order consistent
The central difference approximations of the first and second spatial derivatives are 2nd order consistent
What is Stability
A numerical method is stable if it produces an approximation of the true solution, which exactly solves a “nearby” differential equation
What is the Region of Absolute Stability
What is the region of absolute stability for the explicit Euler method
∣1+λΔt∣< 1
In terms of the complex variable z=λΔt,
Thus the solution Xn converges to zero as n→∞ if and only if λΔt is within this defined circle
What is the region for absolute stability for the implict Euler method
|1 − ∆tλ| > 1
In terms of the complex variable z=λΔt,
Thus the solution Xn converges to zero as n→∞ if and only if λΔt is within this defined circle
What is the advantage of using the implicit Euler method over the the explicit method
This region for implicit Euller that converges is far bigger than the explicit method, and this means that we have a lot more flexibility in our choice of step size ∆t
What is the region for absolute stability for the trapezoidal method
|1 +λ∆t/2|< |1 − λ∆t/2|
What is the Stability Theorem for finite-difference Schemes for BVPs
What do we know about a finite-difference method if the appoximation is both stable and kth order consistent
How is the Method of Lines applied using finite differences to solve the heat equation in one spatial dimension
What is the Crank-Nicolson method
What is the convection-diffusion equation
How can the convection-diffusion equation be solved
Use Central-difference approximation
What is the mesh Peclet number
h|w|/ 2
The centred difference approximation is only stable when the mesh-Peclet number is less than 1
What is the upwind finite-difference method
What does it mean when the upwind finite-difference method is described as “unconditionally stable”
The term “unconditionally stable” in the context of the upwind finite-difference method implies that the numerical scheme remains stable regardless of the value of the grid spacing h, provided h>0
