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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15 views

Numerical scheme to 1D advection equation

I am trying to numerically solve a system of equations which model the early universe in 1D. The equations I am stuck on are; $$ (1)\quad \partial\rho/\partial t + \partial(\rho v)/\partial x = 0 ...
0
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0answers
9 views

First order and second order derivative backwards approximation on non uniform grids

Assume we have a function f(x) and we have samples coming in, except they are not at uniform intervals. (e.g. you have f(x) at x = 0.1, 0.2, 0.7, 1.9, 2.8... etc). What is the correct way to compute ...
0
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0answers
15 views

Determine the order of dissipation for a finite difference scheme

Consider the PDE $u_t + au_x = f$ and the finite difference scheme: $$ \frac{3v_m^{n+1} - 4v_m^n + v_m^{n-1}}{2k} + a\frac{v_{m+1}^{n+1} - v_{m-1}^{n+1}}{2h} = f_m^{n+1} $$ I need to determine the ...
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0answers
32 views

Applying Central Difference (Finite Difference Method) in MATLAB

I was given a rather complicated few problems to solve in MATLAB using the central difference method, and I'd like some help figuring out how to translate this into code. The goal is to discretize ...
0
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1answer
23 views

Matlab solution for non-homogenous heat equation using finite differences

Given the following PDE (non-homogenous heat equation): $$ u_t(x,t) = c^2u_{xx}(x,t) + f(x,t) $$ $$ u(0,t) = u(l,t) = 0 $$ $$ u(x,0) = g(x) $$ $$ 0 < x < l ; t > 0 ; c > 0 $$ I wrote the ...
2
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0answers
28 views

Calculating gradient from finite difference results

I am solving the steady-state heat equation in two dimensions: $$\frac{\partial}{\partial x}\left(k\frac{\partial T}{\partial x}\right) + \frac{\partial}{\partial y}\left(k\frac{\partial T}{\partial ...
3
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1answer
32 views

Bounding the error in the finite difference approximation $\frac{-3f(x) + 4f(x+h) - f(x + 2h)}{2h} - f'(x)$

A course problem asks me, assuming that $f$ is $C^3$ on $\mathbb{R}$ (and $f'''$ is bounded and continuous on $\mathbb{R}$), to show that $$\left| \frac{-3f(x) + 4f(x+h) - f(x + 2h)}{2h} - f'(x) ...
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30 views

What is a reasonable manufactured solution to test finite difference method?

What is a reasonable manufactured solution to test the following equation against its finite difference approximation? I want it to look like a Cosine function about $0$, rotated about $Z$ axis ...
0
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0answers
8 views

Convergence of recursive application of finite-difference operator to $C^{\infty}$ functions

Let $f\colon \mathbb{R}\to \mathbb{R}$ be an arbitrary smooth function (whose extension to a complex differentiable function is entire, if it matters). Let $\mathbf{D}_{h}$ be a finite difference ...
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0answers
26 views

Solving ODE by finite differences and Newton's method.

Given this boundary value problem $y'' = (x^2(y')^2 - 9y^2 + 4x^6)/x^5, \quad 1 \leq x \leq 2, \qquad (1)\\ y(1) = 0, \; y(2) = \ln 256$ I have to solve the problem using finite differences, for 21 ...
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vote
1answer
28 views

3D Finite Difference Matrix

I have been working with a finite difference code for the case in which my problem is axysimmetric. This means that all the code I have so far is for 2D In this case the coefficient matrix isn't ...
0
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1answer
20 views

Fastest numerical way to solve steady-state reaction-diffusion equation

I have a reaction-diffusion equation in 2 dimensions of the typical form: $\frac{\partial u}{\partial t} = D\nabla^2u - \Phi(u(x))$ I want to stress that $\Phi(u(x))$, is not a constant, but depends ...
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0answers
5 views

What is mixed finite difference of function of two arguments?

What is mixed finite difference of function of two arguments? Suppose function is $F(i,j)$ then what is $\Delta_{ij}F$ ? Is it $(F(i+1,j)-F(i,j)) \times (F(i,j+1)-F(i,j))$
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20 views

What does instability mean and examples, boundary condition

The Upwind-Scheme for the numerical solution of first order PDE's (homogenous case) of the form $u_t + cu_x = 0$ is given by $$ u_j^{n+1} = \left\{ \begin{array}{ll} u_j^n - \frac{c\Delta t}{\Delta ...
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0answers
15 views

linear differential operator 2d, order of error h^4?

I have to show that following discretization of a linear differential operator satisfies the equation $\Delta_h u(x) = \Delta u(x) + \frac{h^2}{12} \Delta^2 u(x) + O(h^4)$ $$\Delta_h u = ...
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19 views

Forward-difference approximation using taylor expansion

Consider a Forward-difference approximation for the second derivative of the form, $f''(x) = Af(x) + Bf(x+h) + C(x+2h) $ Use Taylors Theorem to determine coefficients, $ A,B, C$ that give the ...
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1answer
31 views

Finite difference numerical differentiation

I needed to find an O(h2) method to find f'''(x). Using Taylor expansions, I found: $$f'''(x)=\frac{f(x+2h)-2f(x+h)-2f(x-h)+f(x-2h))}{2h^3} + O(h^2)$$. However, I have also found that: ...
0
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0answers
12 views

Finite difference matrix for gradient

I came across this statement: D1 denotes the finite difference matrix mapping the scalar field p to the vector field $\nabla$p. What does D1 represent here? p, I think, is a 2 dimensional field ...
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0answers
17 views

Bouncing back and forth between anti-difference and finite difference in finite calculus at exponential functions.

I've been reading 'concrete mathematics(knuth)' and just don't get how I'm supposed to bounce back and forth between anti-difference and finite difference in finite calculus, specifically at ...
0
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1answer
37 views

ODE and recurrence relation

I am trying to understand the following claim (I came across it while reading a paper): Consider the map (Standard/Arnold map) $T_{k}:(x,y)\mapsto(x+y+kf(x), y+kf(x))$, with ...
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0answers
20 views

Equivalence discrete H^2 Sobolev norms

My aim is showing the equivalence of two discrete Sobolev norms. On $\mathbb{Z}^d$, $d\ge 2$, one defines the discrete derivative in the direction of the coordinate vector $\vec e_j$ as $$ ...
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1answer
61 views

Discretization of a heat equation using finite-difference method

Let $\overline{\Omega}=\left([0,2]\times [0,1]\right)\cup \left([1,2]\times [1,2]\right)$ and $\Gamma=\Gamma_1\cup\ldots\cup\Gamma_6$ be defined as shown in the following picture: I want to ...
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0answers
15 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 ...
0
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2answers
58 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
30 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
40 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 ...
0
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0answers
19 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
16 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
12 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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0answers
36 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
138 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
32 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 ...
2
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0answers
40 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
70 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
votes
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
53 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: ...
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1answer
22 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
votes
0answers
28 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 $e_{i,0}=0$. ...
0
votes
0answers
94 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
votes
0answers
59 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. ...
0
votes
1answer
45 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
36 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 ...
3
votes
4answers
75 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
votes
1answer
66 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. ...
0
votes
1answer
54 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
49 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
33 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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votes
1answer
91 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
140 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
votes
1answer
47 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} } = ...