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.

# Questions tagged [finite-differences]

706 questions

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### Is the derivative the natural logarithm of the left-shift?

(Disclaimer: I'm a high school student, and my knowledge of mathematics extends only to some elementary high school calculus. I don't know if what I'm about to do is valid mathematics.)
I noticed something really neat the other day.
Suppose we…

David Zhang

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### Generalisation of $(n+3)^2-(n+2)^2-(n+1)^2+n^2=4$

After seeing the neat little identity $(n+3)^2-(n+2)^2-(n+1)^2+n^2=4$ somewhere on MSE, I tried generalising this to higher consecutive powers in the form
$\sum_{k=0}^a\epsilon_k(n+k)^p=C$, where $C$ is a constant and $\epsilon_k=\pm1$. I discovered…

Pauly B

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### Has anybody ever considered "full derivative"?

When differentiating we usually take a limit and drop the infinitesimal terms.
But what if not to drop anything?
First, we extend the real numbers with an infinitesimal element $\varepsilon$ which has its own inverse $1/\varepsilon=\omega$.
And…

Anixx

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### What is the difference between Finite Difference Methods, Finite Element Methods and Finite Volume Methods for solving PDEs?

Can you help me explain the basic difference between FDM, FEM and FVM?
What is the best method and why?
Advantage and disadvantage of them?

Anh-Thi DINH

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### Why isn't finite calculus more popular?

I'm reading through Concrete Math, and learning about finite calculus. I've never heard of it anywhere else, and a Google search found very few relevant sources. It seems to me an incredibly powerful tool for evaluating sums, essentially a…

Elliot Gorokhovsky

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### Chain rule for discrete/finite calculus

In the context of discrete calculus, or calculus of finite differences, is there a theorem like the chain rule that can express the finite forward difference of a composition $∆(f\circ g)$ in simplified or otherwise helpful terms?
It's probably not…

GregRos

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### Impose PDE itself as Boundary Condition?

Consider, for example, the elliptic PDE $u_{x}+u_y+u_{xx}+u_{yy}=0$ for $(x,y)\in[0,\infty)^2$.
Solution methods often require me to impose boundary conditions. Often, these arise naturally from applications (physics, biology, economics etc.). But…

Alex

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### Is there a mean value theorem for higher order differences?

The standard mean value theorem tells us $\frac{f(x+h)-f(x)}{h} = f'(c)$ for some $c$ between $x$ and $x+h$. Rewriting this, we may see it as $\frac 1h\Delta_h f(x) = f'(c)$. This makes me wonder if there is a similar formula for higher order…

nullUser

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### Convergence of numerical methods for Viscous Burgers' Equation

For linear problems we can deduce convergence of numerical schemes by checking consistency and stability because of the Lax Equivalence Theorem.
For conservation laws, we know that conservative, consistent and monotone schemes will converge to the…

428

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### What is convection-dominated pde problems?

Can you explain for me what is convection-dominated problems? Definition and examples if possible.
Why don't we can apply standard discretization methods (finite difference, finite element, finite volume methods) for convection-dominated…

Anh-Thi DINH

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### Finite differences second derivative as successive application of the first derivative

The finite difference expressions for the first, second and higher derivatives in the first, second or higher order of accuracy can be easily derived from Taylor's expansions. But, numerically, the successive application of the first derivative, in…

CFDIAC

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### Recurrence relations on a continuous domain

While attempting to read Shannon's paper I came across the following (p. 3): suppose $N\colon \mathbb{R} \to \mathbb{R}$ is a function, which for some fixed (given) set of values $t_1, t_2, \dots, t_n$ satisfies
$$ N(t) = N(t-t_1) + N(t-t_2) + \dots…

ShreevatsaR

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### 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} - u_{i+1,j+1}\right] \quad\quad (1)
\end{align}
Show using…

James

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### Show that the $k$th forward difference of $x^n$ is divisible by $k!$

Define the forward difference operator
$$\Delta f(x) = f(x+1) - f(x)$$
I believe that if $f(x)$ is a polynomial with integer coefficients, $\Delta^k f(x)$ is divisible by k!. By linearity it suffices to consider a single monomial $f(x) = x^n$. …

Geoffrey Irving

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### What is a common framework for these divergent sums?

If you expand $2^x$ using a finite difference series you end up with the formula
$$ 1 + x + \frac{1}{2!}x(x-1) + \frac{1}{3!}x(x-1)(x-2) ... = \sum_{n=0}^{\infty} \frac{(x)_n}{n!} $$
Now these series diverge for negative arguments, but they give…

Sidharth Ghoshal

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