A method in numerical analysis which consists of approximating the derivatives of a solution of an ordinary or a partial differential equation. This leads to the solution of a linear system.

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7 views

Numerical Overflow in Dirichlet Boundary Value Problem On High Dimensional State

I am using multigrid methods to solve a quasilinear parabolic pde with Dirichlet boundaries. The problem is too long to reproduce here, but my question is more practical than theoretical: The state ...
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2answers
44 views

What does the constant mean in Big O notation?

I have a big issue in understanding the real meaning of Big O notation. Classical definition: $f(x) = O(g(x))$ as $x\rightarrow k$ if there exist $\delta, C > 0$ such that $f(x) \leq Cg(x)$ ...
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0answers
19 views

Crank-Nicolson method for solving nonlinear parabolic PDEs

Is the Crank-Nicolson method appropriate for solving a system of nonlinear parabolic PDEs like $\partial u/\partial t - a\Delta u + u^4 = 0$ ? I tried to apply this method for solving such system but ...
2
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1answer
36 views

Solving recurrence -varying coefficient

How can one find a closed form for the following recurrence? $$r_n=a\cdot r_{n-1}+b\cdot (n-1)\cdot r_{n-2}\tag 1$$ (where $a,b,A_0,A_1$ are constants and $r_0=A_0,r_1=A_1$) If the $(n-1)$ was not ...
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0answers
13 views

Discrete Fourier vectors are the eigenvectors for any linear, constant coefficient, periodic, finite difference discretization on a uniform grid?

I came across the following statement: It can be shown that the DF vectors are always the complete set of eigenvectors of any linear, constant coefficient, periodic, finite difference discretization ...
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1answer
12 views

Can anyone explain how to show the finite difference equation $y'_{0}=\frac{y_{1}-y_{-1}}{2h} + O(h^{2}).$?

I was given that $y_{j}=y(x_{j})$ where $x_{j}=x_{0}+jh$ for integer j and positive h. I need to show that $$y'_{0}=\frac{y_{1}-y_{-1}}{2h} + O(h^{2}).$$ I thought I could start by finding the Taylor ...
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0answers
11 views

Question About FDM and Boundary Conditions

I'm taking a numerical analysis class and have a quick question about boundary conditions. I don't understand how boundary conditions are incorporated into the matrix form of the FDM. Can someone ...
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22 views

Finite difference method for nonlinear partial differential equations

I have the following partial differential equation (PDE) $ \forall (x,t)\in(0,L)\times(0,\infty) $ \begin{equation} \begin{split} m_{z}\ddot{w}&+EIw'''-Tw''-f+c_{1}\dot{w}-EAv''w'-EAv'w'' ...
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2answers
133 views

Wikipedia wrong? Convergence of finite difference

Update: I have edited the Wikipedia page, so that the mistake no longer appears. On the Wikipedia article for "Finite difference" there is the claim Assuming that $f$ is continuously ...
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0answers
22 views

Second derivative approximation at the endpoint of a bounded function

I have a function defined on [a, b] and trying to approximate its second derivative using finite differences method. The centered finite difference formula works for interior points, but not for ...
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0answers
31 views

Boundary Conditions for a Finite Difference Approximation of a Sixth Derivative

I am trying to use a finite difference scheme to numerically solve sixth order parabolic equations such as \begin{equation} u_t = u_{xxxxxx} \end{equation} with symmetry conditions \begin{equation} ...
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1answer
67 views

Second difference $ \to 0$ everywhere $ \implies f $ linear

Exercise 20-27 in Spivak's Calculus, 4th ed., asks us to show that if $f$ is a continuous function on $[a,b]$ that has $$ \lim_{h\to 0}\frac{f(x+h)+f(x-h)-2f(x)}{h^2}=0\,\,\,\text{for all }x, $$ then ...
2
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0answers
74 views

What does this expression give?

It is known that $$f(x+n)=\sum_{k=0}^n {n\choose k} \Delta^k[f](x)$$ Also it is known that $$\Delta^n [f](x)= \sum_{k=0}^n {n \choose k} (-1)^{n-k} f(x+k)$$ Now I encountered the following ...
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0answers
36 views

stability of FTCS scheme for parabolic equation

Can you suggest any method for stability analysis of FTCS scheme for the the following parabolic equation ? D.E: $u_{t}=a(x,t)u_{xx}+f(x,t,u)$, $0<x<1$, $0<t<T$, $T>0$ BCs: ...
1
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1answer
20 views

Discretization of v*(du/dx)

I am trying to discretize the term: $$\underline{v}\frac{d\underline{u}}{dx}$$ using finite differences or evaluate $$\int_{\Gamma}\underline{v}\frac{d\underline{u}}{dx}.\underline{n}d\Gamma$$ ...
2
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0answers
24 views

Solution of partial difference equation

I want to find the explicit solution of the following difference equation $e_{i,j+1}=re_{i-1,j}+(1-2r)e_{i,j}+re_{i+1,j}+km_{i,j}$ where $r>0$, $k>0$ and $m_{i,j}$ are known and $e_{i,0}=0$. ...
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0answers
53 views

Finite Difference - Forward Difference with 2nd order Accuracy: What to do at the boundary?

I implemented a BVP using a first-order finite difference scheme after the shooting method did not work reliably. Its the first time I have worked with this. The code works but I would like to move to ...
2
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0answers
55 views

How to solve this finite-difference equation?

How to solve the following finite-differences equation: $$f(x) = f(x-1) + f(x-\sqrt{2}), \quad x\in [\sqrt{2}, +\infty) \,?$$ Let's say $f(x) = f_0(x)$ for $x \in [0, \sqrt{2})$ is a given function. ...
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1answer
41 views

How to discretize mixed partial derivatives?

How to discretize $\frac{\partial^3 f}{\partial x\partial y^2}$ at mesh point $(i,j)$? We should use mesh points which are nearest to $(i,j)$.
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1answer
34 views

$\Delta^kn^\alpha$ converges monotonically to zero when $\alpha<k$

I am trying to prove that the $k$-th finite difference of the series $n^\alpha$ converges to zero monotonically as $n\to\infty$ when $\alpha<k$. The differential analogue is ...
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4answers
73 views

Elementary Proof of sum of alternating factorials

so I came across the rather interesting identity $$ \sum_{i = 0}^{\infty}\left[\sum_{j=0}^{i}\left[\frac{(-1)^j}{j!} \right] \prod_{j=0}^{i}\left[ (x-j)\right] \right] = x! $$ For positive integers ...
2
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1answer
56 views

Checking if the Hessian is the derivative of the gradient

Suppose $f: \Bbb R^n \to \Bbb R$. I have a code that computes the gradient of $f$. I have another code that computes the Hessian of $f$ times a vector. Now I want to check if they are correct. ...
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1answer
49 views

Solving ODE with matrices

I have an equation in ODE $M{'}(x)= M(x)*A(x)$. Issue here is $A(x) = C_1+C_2* x $ where $C_1,C_2 $ has dimension $3 \times 3$. And x is a scalar variable Doubt What is M(x)? Can any one give ...
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1answer
42 views

Finite Difference Method Stability with diffusion equation

The diffusion equation is: $ \frac{\partial T}{\partial t} = \alpha \left( \frac{\partial^2 T}{\partial x^2} \right) $ An explicit finite difference approach can be used to solve this, forward in ...
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0answers
30 views

Space-dependent diffusivity and finite-differences

I want to implement a finite difference code of this simple diffusion equation with space-dependent diffusivity: $$\partial_{t}u =D\partial_{x}^{2}u+\partial_{x}D\cdot\partial_{x}u$$ I go for a ...
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1answer
66 views

Second derivative of discrete function

Given function $y[n]$ what is the best way to define the second derivative? Some background to the question: in linear systems we often sample a continuous signal to a discrete one with sample rate ...
2
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0answers
124 views

Finite Difference Spacing of Points for PDE's for Convergence of Explicit Forward-Stepping Scheme

I realize that this question could be pretty broad, but I'm wondering at least what the conditions are for my simulation. I'm developing an Explicit Forward-Stepping Finite Difference scheme to solve ...
2
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1answer
42 views

Finite differences to ODE in polar coordinates

I have an equation of finite differences as follows: $$\frac{X_1(r+\epsilon)-X_1(r)}{ \frac{\epsilon~\beta}{2~r+\epsilon} } + \frac{X_1(r-\epsilon)-X_1(r)}{ \frac{\epsilon~\beta}{2~r-\epsilon} } = ...
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2answers
44 views

Avoiding substraction for finite difference with log and exp

I want to approximate the derivative of f(x) Finite difference $f'(x) \approx \frac{f(x+h)-f(x)}{h}$ I was taught that the error from the substraction is blown up for small h. This I can verify ...
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0answers
31 views

heat equation with Interface Crank Nicolson

I am currently working on solving the heat equation with an interface numerically using Crank-Nicolson. There are jump discontinuities at the interface which are dealt with using fictitious values ...
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0answers
35 views

Stability conditions

Below is a problem about stability conditions that I have been struggling with it during an exam: Find the stability conditions for $$A\left ( \frac{\partial^2 u(x,\, y,\,t)}{\partial x^2} + ...
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0answers
14 views

Why more smooth the function the more precise finite difference method?

As the title, Why more smooth the function the better finite difference method? I guess that if the function is smooth we can better approximate with Taylor series, but formally how this helps? ...
2
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0answers
107 views

Boundary integral method to solve Poisson equation

Suggest how to solve Poisson equation \begin{equation} σ ∇^2 V = - I δ(x-x_s) δ(y-y_s) δ(z-z_s) \nonumber \end{equation} by using the boundary integration method to calculate the potential ...
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1answer
39 views

Finite differences coefficients

I'm interested in deriving a forward finite difference approximation for the gradient of a function, $f(x)$, at the point $x = x_i$ using $k+1$ points. If the spatial domain is uniformly discretized, ...
4
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1answer
186 views

9 point stencil for Laplacian operator

Given the following 9 point Laplacian \begin{align} -\nabla^2u_{i,j} = \frac{2}{3h^2}\left[5u_{i,j} - u_{i-1,j} - u_{i+1,j} - u_{i,j-1} - u_{i,j+1} - u_{i-1,j-1} - u_{i-1,j+1} - u_{i+1,j-1} - ...
5
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1answer
90 views

Who introduced the finite difference notation using $\Delta$?

We all know that Leibniz introduced the differential notation $dx, dy$, and that in developing his calculus for infinitesimal differences he was inspired by his previous work on finite diffences. ...
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0answers
12 views

Why is it that if $||\textbf{v}^{n+1}||_h \leq (1+kK)||\textbf{v}^{n}||_h$ then the scheme is stable?

Question I'm having difficulty understanding a lemma in my notes. The lemma is: If there exists a constant K, independent of $h,k,n,\textbf{v}^0$, such that $||\textbf{v}^{n+1}||_h \leq ...
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29 views

Finite Differences and Scale Invariance

The semilinear heat equations $u_t = u_{xx} + u^p$ is invariant under the one parameter family of scalings $ t \to \lambda t, \quad x \to \lambda^{1/2}x, \quad u \to \lambda^{-1/(p-1)}u$. When we ...
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1answer
29 views

Finding the value of Cmax for the Courant–Friedrichs–Lewy condition

According to the Courant–Friedrichs–Lewy condition, in the 1 dimensional case we have: $C=\frac{u\cdot\Delta_t}{\Delta_x} \leq C_{max} $ It is said that $C_{max}$ changes depending on the method ...
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0answers
40 views

Finite Difference Scheme for a PDE on non-rectilinear coordinates

Consider poisson's equation on the domain $0 \leq x \leq 1$ $0 \leq y \leq H(x)$. Change the coordinates to $\xi=x$, $\eta=y/H(x)$. Construct a FDS that gives a positive definite symmetric matrix. ...
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1answer
28 views

Raising the power of forward difference formulas

In a forward difference formula, $D_-$ refers to the backward difference operator. Thus, $(D_- u)(x) = u(x)-u(x-h)$. In the answer key of a problem that I was working on, one of the steps is: ...
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1answer
55 views

General solution of a finite-difference equation with real non-commensurable differences.

How to find a general solution of the following functional (recurrence) equation: $$f(x) = c_1 f(x - a_1) + \dots + c_n f(x - a_n), \tag{1}$$ where $c_i, a_i$, $i = 1, ... n$ are arbitrary real ...
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0answers
28 views

Discretization 5 equations and unknowns on a 2d grid

I would like to solve the following set of equations using a finite difference approach with a 2D-grid. $\frac {Dφ}{Dt}$ = (1 − φ)P + Γ with $\frac {Dφ}{Dt}$ = $\frac {\partialφ}{\partial t}$ + V · ...
3
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0answers
48 views

Is there a special name for pi-based finite difference?

Did anybody consider $\pi$-based finete differences, that is the operator $$\Delta_\pi f(x)=f(x+\pi)-f(x)$$ and its corresponding inverse operator? It seems for me that taking the step equal to $\pi$ ...
3
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1answer
120 views

Having trouble understanding the finite element method

I am trying to read some sources on how to implement the finite element method and I am having difficulty putting all the concepts together. I can read and understand the Galerkin approach just fine. ...
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44 views

Finite difference discretization of the Porous Medium Equation - How to prove stability?

The porous medium equation is given by $\delta_t u(x,t) = (u^p \cdot u_x)_x, (x,t) \in \mathbb{R} \times (0, \infty)$ We are working on approximating the solution with finite differences. ...
4
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1answer
31 views

Numerical computation of the $n^{\mathrm {th}}$ derivative of a multivariate function

From a multivariate function $f$, depending on $n\geq 1$ variables, which can be computed numerically, but which does not admit simple analytic expression, I would like to approximate numerically the ...
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0answers
106 views

FDTD validation with Poynting Vector

I'm attempting to validate my FDTD optics simulator results. Visually, I can see that my output is nearly identical to that produced by Meep, the only difference being a slight phase shift. Using a ...
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0answers
72 views

Calculate a 5x5 Vandermonde system for a 5 point mesh

This is problem 1.2 in Randall J Leveque's textbook, "Finite Di fference Methods for Ordinary and Partial Di fferential Equations". I'm struggling with how to actually do the computation, I'm not so ...
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0answers
15 views

finding most and least they can afford

The Johnsons have accumulated a nest egg of $30,000$ that they intend to use as a down payment toward the purchase of a new house. Because their present gross income has placed them in a relatively ...