Questions tagged [finite-differences]
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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Express eigen values of Gauss Seidel matrix?
In matrix form the Gauss Seidel iterations can be expressed as:
$$X^{m+1} = (D + L)^{-1}\textbf{b} - (D + L)^{-1}UX^{m}$$
Where $A = D + L +U$ is a finite difference matrix expressing the original ...
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Where did I make a mistake while trying to derive the CFL condition using the matrix norm method
Using the central difference derivative approximation, the finite difference approximation of the following wave equation:
$$ c^2 u_{xx} = u_{tt} \tag 1$$
can be written as:
$$ U_{i+1, j}= kU_{i, j+1} ...
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Deriving Finite Difference Scheme for Goursat PDE on Triangular Domain
I have the following PDE:
$k_{xx}(x, y) - k_{yy}(x, y) = \lambda(y)k(x, y)$
$k(x, 0) = 0$
$k(x, x) = -\frac{1}{2} \int_0^x \lambda(y) dy$
defined on the triangle $0 < y \leq x \leq 1$. ($k_{xx}$ is ...
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Boundary Conditions for 4th order PDE in Matrix form with Finite Difference
I have a PDE in the form:
$$ \frac{\partial^4 y}{\partial x^4} = 10 $$
with boundary conditions
$$ y(0,t) = 0 $$
$$ \frac{\partial y}{\partial x}(0,t) = 0 = \frac{\partial^2 y}{\partial x^2}(1,t) = \...
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Stability of finite difference scheme by bounding a norm
This question comes from Strikwerda, "Finite Difference Schemes and Partial Differential Equations":
By multiplying
$$
\frac{v^{n+1}_m - v^{n}_m}{k} + \frac{a}{2}\left( \frac{v^{n+1}_{m+1}-...
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How to "numerically" calculate eigenvalues of differential operator using Finite difference?
Consider a simple problem $$ \frac{d^2y}{dx^2} y = -\lambda^2 y.$$ I would like to calculate the eigenvalues of operator $\, d_x^2\, $ using finite difference, but I am not sure how to do it. ...
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Numerically solving a BVP for a 2nd order ODE with mixed Dirichlet and Neumann boundary conditions with the finite difference method
I would like to solve a simple BVP for a second order differential equation on the domain $I=[0,\pi/2]$ using the finite difference method.
$$
u''(x)+u(x)=0\\
u(0)=0\\
u'(\pi/2)=0
$$
By construction, $...
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Implication about the existence of f'(x) [duplicate]
Let $h>0$. Consider the finite difference formula
$$
\frac{f(x+h)-f(x-h)}{2h} \approx f'(x)
$$
which is called central difference. It is well known that for a sufficiently smooth function $f(x)$,
$$...
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How is this finite difference expression derived? [closed]
In this $\text{paper}^1$, the following equation:
$$\frac{\partial O}{\partial t}(p,x;t) = \left[p\frac{\partial}{\partial x} - \left(\gamma p + \frac{dv(x)}{dx} \right)\frac{\partial}{\partial p} + \...
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Finite difference scheme for mixed time and space derivatives
I have this PDE
$\frac{\partial^2 u}{\partial x^2}+\frac{1}{x}\frac{\partial u}{\partial x}+\frac{\partial }{\partial t}(\frac{\partial^2 u}{\partial x^2}+\frac{1}{x}\frac{\partial u}{\partial x})-\...
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Can I use this condition to solve a radially summetric equation with the finite difference method
Consider the symmetric radial heat equation:
$$ f_t = \nabla^2f \Leftrightarrow f_t = f_{rr} + {1 \over r}f_r \tag 1$$
To solve this with the finite difference method using the FTCS scheme, we need ...
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Wave equation solved by MOL - stability condition for 5-point central formula
I want to derive the stability condition for the wave equation (5-point central finite difference formula).
Wave equation 1D is defined as:
$$
\frac{\partial^2 u}{\partial t^2} = \alpha\frac{\partial^...
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Ghost point vs. three-point one-sided differencing in a Poisson-like PDE
I am trying to numerically solve the following PDE for $u\left( z \right)$:
$$ \frac{\partial}{\partial z} \left( \eta \frac{\partial u}{\partial z} = c\right), $$
$$ \eta = a \left( \frac{\partial u}{...
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1
answer
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Why is the Lax-Wendroff Finite Difference scheme 2nd order in time and space?
I'm struggling to understand why the Lax-Wendroff scheme is second-order in time and space.
My derivation of the scheme for the linear wave equations is as follows:
\begin{equation}
u_{t} +cu_x = 0
\...
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1
answer
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High-precision second-order difference quotient of 2 variables functions
For one variable function, 2th order difference quotient can be:
$$
\frac{f(x+h) - 2f(x) + f(x-h)}{h^2},
$$
which can also be:
$$
\frac{-f(x-2h) + 16f(x-h) - 30f(x) + 16f(x+h) - f(x+2h)}{12h^2}
$$
...
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BVP for $y'' - 2y^3 = 0$ where $h= 0.25 , y(-1) = 0.5 y(0) = 0.5 , [-1.0]$ by using finite difference method
find BVP for $y'' - 2y^3 = 0$ where $h= 0.25 , y(-1) = 0.5 y(0) = 0.5 , [-1.0]$ by using finite difference method.
I have been trying to solve this but I got a system of equations where my variables ...
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Finite Difference Method for BVP with Robins Condition
If you have a nonlinear second-order boundary value problem where the domain of the problem is $x \in [a,b]$, the boundary conditions imposed are the Robins condition (mixture of the dependent ...
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How to bypass singularity point at $r = 0$ using Maclaurin's expansion for this symmetric problem?
Consider the following time-radial partial differential equation:
$$U_t(t,r) = \nabla^2 U(t,r) = U_{rr}(t,r) + {1\over r}U_r(t,r) \tag 1$$
If we would want to solve the above PDE numerically by using ...
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Residual for the derivative at the boundary (Relaxation method for ODEs)
I'm trying to solve a second-order boundary ODE using the Relaxation method for ODEs.
Given a boundary ODE,
$$\frac{d^2z}{dx^2} = g(x,z,z'), \quad z(a) = \alpha, \quad z(b) = \beta, \quad x \in [a,b]$$...
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Numerical solution of PDE unstable
I'd like to solve the heat equation numerically:
\begin{equation}
\partial_tu=\partial_x^2u,
\end{equation}
and I've tested my algorithm with initial conditions $u(0,x)\propto\operatorname{exp}[-(x-...
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How does this finite difference local truncation error behave when changing its parameters and how to find its boundary
I derived the local truncation error of the following partial differential equation:
$$ f_{xxxx}-f_{xxx}+{1 \over 4}f_{xx}+e^x f_{zzzz} + \beta e^x f_{tttt} +2 e^{x/2}f_{xxzz} - \alpha e^{x/2}f_{xxtt} ...
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Matrix method stability analysis of the numerical finite difference $2$D heat equation problem
I am trying to determine the finite difference method stability condition of the following $2$D heat equation by using the matrix method analysis instead of Von Neumann analysis:
$$ u_{t}=q(u_{xx} + ...
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A finite difference method for a nonlinear system of PDEs
I've been trying to write an explicit FDM for this nonlinear system of PDEs:
$$ \frac{\partial u}{\partial t} = f(u, v) + \frac{\partial ^2 u}{\partial {x}^2} - \frac{\partial}{\partial x} \left(u \...
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What is three-point backward difference?
I fully understand two-point, forward/backward/central difference. I know them mathematically and graphically but I'm not quite sure what the three-point backward difference is. I know that we get the ...
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Compact Finite Differences for the Heat Equation with Robin Boundary Conditions
I am trying to solve the Heat equation with Robin Boundary condition:
$$ u_t(x,t) = u_{xx}(x,t), \\ u(x,0) = g(x), \\ u(0,t) + u_x(0,t) = h_0(t), \\ u(1,t) + u_x(1,t) = h_1(t)$$
for $ 0\leq x\leq1$ ...
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Predictor-Corrector Method (FDM, FVM)
I was searching to find an example predictor-corrector solving method applied either to FVM or FDM scheme regarding either fluid or heat flow.
Do you have any ideas where I can find some examples?
...
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Von Neumann stability for inhomogeneous PDE
I've got an inhomogeneous PDE of the following form:
$$\alpha\partial^2_xu+\partial_tu=f$$
with $\alpha<0$ and a source term $f$. I descretise $u$ according to $u_{m,n}=u(m\Delta t,n\Delta x)$ ($f$ ...
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Application of Crank-Nicholson method to higher order time derivatives
In literature, the Crank-Nicholson method is always used to solve only the heat equation. The heat equation contains a first order time derivative. I never read in any literature that the Crank-...
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What is a sufficient condition for stable convergence of a variable coefficient partial differential equation solution solved by the FDM?
While studying the finite difference method for solving partial differential equations, I came across this chapter of a book. On the first paragraph of page $2.27$ it says that the Von Neumann ...
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Literature request on the subject of finite difference method for solving PDEs with variable coefficients
I am trying to understand how to solve PDEs with variable coefficients using the finite difference method. I read a few books about numerical solutions to PDEs, but none discuss how to solve a PDE ...
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PDE numerical stability condition can not be determined by using the Von Neumann stability analysis
The Von Neumann stability analysis is used to determine the stability conditions of finite difference schemes. However, the functions that are contained in the PDE are restricted to only two ...
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Order of a numerical iteration method
Suppose that we have the definition of order $p$ of a numerical method as in the first snippet below. Now I want to prove that for a one-point iterative method this order $p$ is a positive integer. To ...
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Finite difference method solves the wave equation for one set of bounday conditions, but does not when I change them
I wrote a finite difference algorithm in Matlab to solve the wave equation which is derived here.
When I ran my code, the plotted graphs of the numerical and analytical solution deviated, which is the ...
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asymptotics of finite difference equations
I am looking at problem B4 of The 73rd William Lowell Putnam Mathematical Competition Saturday, December 1, 2012:
What is the asymptotics of the following difference equation?
$$a_{n+1}=a_n+e^{-a_n}, ...
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Courant–Friedrichs–Lewy condition for planar FTDT
Courant–Friedrichs–Lewy condition in FDTD is used to define minimal time step to remain stability of the solution.
The definition of Courant–Friedrichs–Lewy condition in N-dimensional space says, that
...
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Discretization of the 1D Laplace operator with von Neumann boundary conditions
Consider the function on interval $[0, \pi]$ subject to homogenous von Neumann conditions $f'(0) = f'(\pi) = 0$:
$$\Delta f = \lambda f $$
With $\Delta$ is the 1D Laplace operator and $\lambda$ the ...
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fixing the error to be the same in every point in grid
maximum norm of the error is smaller than 2% at every point in the grid. That is, we
measure the relative error against the true solution u
and wish to choose the largest $h_{t}$ and $h_{x}$ (say, up ...
user1119555
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Reporting rate of convergence in numerical experiments
I've recently run a numerical experiment and attempted to report the rate of convergence and my advisor has told me it's wrong. I cannot understand what is wrong about it. I'm hoping someone can take ...
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Solving a linear system of equations (finite difference method). Treatment of ghost cells?
I am trying to solve the Poisson equation on a 3D grid:
$\dfrac{\partial^2 f}{\partial x^2}+\dfrac{\partial^2 f}{\partial y^2}+\dfrac{\partial^2 f}{\partial z^2}=C$
The equation is discretized using ...
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Advection Equation in 2D with Variable Coefficient
I want to solve the equation:
$$ u_t +\partial_x\left(e^{-x}u \right) = 0$$
with the BC $u(0,z,t)=u_0$ on the domain $D=[0,10]\times[0,1]$. My attempt is to use the following approximations:
$$ \frac{...
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Most efficient way to solve a 2nd-order parabolic partial differential equation using finite difference methods.
I have the following 3-dimensional 2nd-order parabolic partial differential equation that I need to solve using the finite difference method:
$\dfrac{\partial^2 f}{\partial x^2}+\dfrac{\partial^2 f}{\...
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Abel's Summation Formula with Discrete Calculus
I would like to prove the following version of Abel summation
$$\sum_{k=1}^nu_kv_k = u_{n+1}\sum_{k=1}^nv_k - \sum_{k=1}^n\Delta u_k\left[ \sum_{p=1}^kv_p\right]$$ where $\Delta u_k = u_{k+1} - u_k.$...
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Finite difference formula for edge detection
I have to apply finite difference method for edge detection
formula for finite difference - $f'(x_i) = \frac{f(x_i+1)-f(x)}{h}$
how do i apply this to a image (represented by matrix) to detect the ...
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1
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Solving ODE with derivative boundary condition with finite difference method by central approximation
I am trying to solve the following ODE:
$$ \frac{d^2y}{dx^2}=y(x) $$
Where I have two boundary conditions: $ y(0)=10 $; and $ \frac{dy(x\rightarrow\inf)}{dx}=0 $
I am trying to solve the problem ...
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2
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Leibniz' Rule for Finite Differences
I am attempting to prove a higher-order product rule in discrete calculus:
I'd like to show $$\Delta^n [f(x)g(x)] = \sum_{k=0}^n \binom{n}{k} (\Delta^kf(x))(\Delta^{n-k}T^kg(x))$$ where $\Delta f(x):= ...
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How do you apply a Butcher tableau for an implicit method with matrices?
I'm a little bit confused about how to understand Butcher tableaus for implicit methods, when I have a matrix. Say I have some ODE that is defined by
$$
\frac{\partial \mathbf{y}}{\partial t} = \...
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Convert difference equation to differential equation
Suppose I have the following third-order difference equation:
Y[t]+aY[t-1]+bY[t-2]+cY[t-3]=G d^t
I need to convert it into a differential equation and I followed this method:
y' = y(t+1)-y(t);
y'' = y'...
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Under which conditions does (v-w)>(x-y) imply (v*a-w*b)>(x*c-y*d)?
I know that
$(v-w)>(x-y)$
$(v-w), (x-y) > 0 $
$(a-b)<(c-d)$
$(a-b), (c-d)<0$
$v,w,x,y,a,b,c,d>0$
Can I conclude from this that $(v*a-w*b)>(x*c-y*d)$?
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Finite Difference discretization of squared transient term
If we were to discretize with Finite Difference the following transient term:
$$\frac{\partial u^2}{\partial t},$$
would it be okay to write
$$\frac{(u^{j+1})^2-(u^j)^2}{\Delta t}.$$
Or does this not ...
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Calculating the numerical Hessian from an elevated Delaunay triangulation
Suppose I have an elevated Delaunay triangulation such as below:
Suppose the vertices are embedded as $(x_i,y_i,z_i) \in \mathbb{R}^3$ where $i \in \{1, \cdots, m \}$.
Let us assume that $z(x,y)$ is ...
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