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]
815 questions
4
votes
2 answers
Discretize second order derivative using the finite difference scheme
I have some problems in Discretize the second order derivative of this equation
after I find u(x) by integrating it, I have problem in discretize it ,, I'm not mathematical person and I try to read a lot .
any help ?!
user1483768
- 43
4
votes
0 answers
Stability of a scheme for one way wave equation.
Multiplying the scheme
\begin{equation*}
\dfrac{v_m^{n+1} - v_m^n}{k} + \dfrac{a}{2}\left(\dfrac{v_{m+1}^{n+1} - v_m^{n+1}}{h} + \dfrac{v_m^n - v_{m-1}^n}{h}\right) = 0
\end{equation*}
by $v_m^{n+1} + v_m^n$ and summing over all values of $m$,…
c.yilmaz
- 61
3
votes
0 answers
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 $\pi$ simplifies many identities much.
For…
Anixx
- 9,119
3
votes
1 answer
Consistency versus convergence
In regards to approximating partial differential equations with finite difference, is there a difference between consistency and convergence? Because I am not seeing it.
Consistent scheme : Discrete operator converges to the PDE operator as the mesh…
Frank
- 880
3
votes
1 answer
$(-1)^t$ in Difference Calculus
I'm interested in finding sums of the form
$\sum_i{\sum_j{(-1)^{N-i+j}} }$
where $N$ can be supposed to be constant.
I'm using the difference calculus to help solve these sums efficiently. I believe that I can simulate this sum…
Matt Groff
- 6,117
3
votes
3 answers
Finite differencing of the diffusion (heat) equation
I am attempting to code a problem for a meteorology class. Our initial equation was as follows:
$$\tfrac{\partial u}{\partial t} = \nu \tfrac{\partial^2 u}{\partial x^2} (*)$$
We were then assigned this finite differencing to…
Alex
- 33
2
votes
1 answer
Regarding Crank-Nicolson Scheme functions on the LHS
Assuming I have a differential equation
$$
\frac{\partial u}{\partial t} = \frac{1}{f(x,t)} F(u,x,t)
$$
The Crank-Nicolson scheme would have the equation discretized as such
$$
\Big[ \frac{\partial u}{\partial t} \Big]_{i+1/2,j} = \frac{1}{2}…
David G.
- 260
- 1
- 11
2
votes
1 answer
Finite difference central scheme equivalent for non centered point
I need to use a central difference scheme:
$$\frac{dy}{dx}=\frac{y_{i+1}-y_{i-1}}{2 \Delta x}$$
equivalent for non centered point:
Unfortunately I cannot find a book in which I have this formula. Could anybody give me this in answer?
Misery
- 571
- 2
- 8
- 23
1
vote
1 answer
Finding the value of Cmax for the Courant–Friedrichs–Lewy condition
According to the Courant–Friedrichs–Lewy condition, in the 1 dimensional case we have:
$C=\frac{u\cdot\Delta_t}{\Delta_x} \leq C_{max} $
It is said that $C_{max}$ changes depending on the method used, but for explicit methods it should be…
Twilight Sparkle
- 191
1
vote
0 answers
The last finite difference of $P$ considering in a general form
Problem: Consider a polynomial (monic) $P$ and finite differences as $Q(x):= P\left(x+\tfrac{1}{2n}\right)-P(x)$ and so on, for any real $a$. Show that the sum $$\sum_{k=1}^n (-1)^k \dbinom{n}{k} P\left(x+\frac{1}{2n} \cdot…
1
vote
0 answers
Handling a discontinuous grid in finite difference
I am supposed to model heat transfer through a cavity that absorbs concentrated solar radiation. The discretization of the cavity is sort of set by a previous ray-tracing model. I end up with something like in the attached image, where two plates…
F.Salih
- 11
1
vote
0 answers
Two-Dimensional Steady-State Heat Flux
Thank you in advance for any help with this assignment.
I promise, I have spent hours trying various sources to understand these terms and operators, but I just cannot make the fundamental leap necessary to know what's going on.
The problem…
Shaun Love
- 11
1
vote
0 answers
$f(x)=e^x$ Prove $f[x_0,x_1,...,x_n]>0$
The function is given that
$$f(x)=e^x$$
Prove that the divided difference is positive for any distinct numbers $x_0,...x_n$ for $n\geq0$. First divided difference is
$${ f[x_1]-f[x_0]\over x_1-x_0}$$ Now if $x_0>x_1$
$${ e^{x_1}-e^{x_0}\over…
Don
- 636
1
vote
1 answer
Choose the best discretization for Finite Differences
I have the following problem:
$$ -u''(x)+5u'(x)=f(x), x \in (0,1) $$
$$ u(0)=u(1)=0 $$
Now I have to find a discretization for Finite Differences, so my Matrix $A_h$ is strictly diagonally dominant (Where $A_h \cdot u=b_h$ ).
How do I know which…
ChrizZly
- 35
1
vote
1 answer
Geometrically Defining Discrete Integral
We define the forward difference as an operator on real (or complex) functions as $D[f] = f(x+1) - f(x)$
It follows then that there is a forward anti-difference that can be defined, which we'll denote as
$${\Large{\mathfrak{D}}}_{a}^{b}[f] …
Sidharth Ghoshal
- 16,771