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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
18 views

Kronecker product and finite difference discretization for poisson equation

In this notebook from MIT's Intro to Linear PDEs course, it is unclear to me why the Kronecker product is used to formulate the coefficients matrix $A$ for solving the linear system of equations $A u =...
Jared Frazier's user avatar
1 vote
1 answer
27 views

How do we find the weights for finite differences?

My professor gave us this question to solve, but I don't know have much familiarity with the topic. Question The forward difference is first order accurate and is defined to be $$ D_{+} = \frac{f(x+h) ...
Calum's user avatar
  • 305
1 vote
0 answers
37 views

Convergence of "partial" PDE finite difference scheme

Suppose I have a PDE $$ \frac\partial{\partial t}f(x,t) = F\left(f(x,t), \frac\partial{\partial x} f(x,t)\right). $$ I would like to discretize $f(x,t)$ as $$f^\epsilon(x,t) = \sum_{n=0}^{N_\epsilon} ...
Thorstein's user avatar
1 vote
0 answers
15 views

Weak convergence of derivatives - Backward Differentiation Formula - BDF 2 method

I am using a space-time discretization to discretize an initial value problem, using P1-FEM for space discretization and BDF-2 for time discretization. To fix the notations: $\Omega\subsetneq\mathbb{R}...
GCMA's user avatar
  • 31
2 votes
2 answers
62 views

Final value of a recursion

Problem Given $p_1, \sigma > 0$, consider the following recursion \begin{equation*} p_{i}=(1-L_i)p_{i-1} \qquad i=2,\dots,k \end{equation*} where \begin{equation*} L_i \triangleq \frac{p_{i-1}}{p_{...
matteogost's user avatar
0 votes
0 answers
50 views

Numerical Analysis of Differential Equation with Boundary Conditions

Note I have decided to edit this post so it is more specific and also because I was incorrect the first time. I didn't wanna make another post about the same subject, but if this is not allowed then ...
Need_MathHelp's user avatar
0 votes
0 answers
7 views

implicite scheme for reaction diffusion equations under finite difference

I am trying to implement an implicit scheme for the following RD equation: $$ u_t = \Delta u+f(v)u $$ where $v$ is a known data and $f$ is the reaction function. The initial is zero and non flux. To ...
79999's user avatar
  • 137
-1 votes
0 answers
23 views

How do I Helmholtz decompose a vector field on a two dimensional lattice if the curl and divergence of the said field is known? [closed]

I have a 2d vector field obtained from processing an image. I also have at my disposal the divergence D and curl C of this field. I want to Helmholtz decompose this field to obtain its rotational and ...
Pranav Kumar's user avatar
0 votes
0 answers
19 views

Numerical method for a free boundary problem

I am consider the following free boundary problem: $u_t + \mathcal{L}u = 0, 0<x<s(t)$, with the boundary conditions: $u(T,x) = f_T(x)$, $u(t,s(t)) = g(t)$ and $u_x(t,s(t))=C$. Here, $\mathcal{L}...
Kenneth Ng's user avatar
-2 votes
0 answers
25 views

Why is the finite difference equation for the diffusion problem always unstable if centered difference is used for both t and x term?

From Walter A. Strauss Partial Differential Equations: An Introduction Chapter 8 Question 10. For the diffusion equation $u_t = u_{xx}$, use centered differences for both $u_t$ and $u_{xx}$. (b) Show ...
xysheep's user avatar
0 votes
1 answer
21 views

How would you Illustrate the Symplectic Euler method for a general initial value problem of the second order to a person who has never heard of it? [closed]

Consider a person who has never heard of symplectic Euler but knows about Initial Value Problems, forward Euler, backward Euler, and Taylor series. Say, you are required to give them an idea of what ...
rayyan's user avatar
  • 9
0 votes
0 answers
28 views

Convergence of Finite Difference Scheme for Helmholtz equation

Consider the 2-dimensional Helmholtz equation $$-\Delta u-k^2 u= f.$$ I'm trying to proof convergence of the standard finite difference scheme (using the 5-point-stencil for the Laplacian). I have ...
Ghost's user avatar
  • 1
1 vote
1 answer
32 views

Spectrum of discretized Laplace operator with homogeneous Neumann B.C.s

Consider the Laplace operator over a 1D domain with homogeneous Neumann B.C.s. The cell-centered Finite Volumes discretization of this operator on a uniform grid looks as follows, which is a well-...
Nicola's user avatar
  • 51
0 votes
0 answers
46 views

How to solve this coupled PDE eigenvalue problem numericallly?

I'm solving a system of two coupled partial integro-differential equations for two functions $ {\phi _0^\alpha (R,r')} $ and $ {\phi _0^\beta (R,r)} $: $$ \frac{{{\partial ^2}\phi _0^\alpha }}{{\...
HERMIT's user avatar
  • 1
1 vote
0 answers
95 views

Do higher order differences within explicit schemes negatively effect stability?

I was reading the paragraph in section E.4 of this paper, where the authors discuss adding longer histories in neural/numerical PDE solvers and claim that "using higher-order differences [in time]...
user572780's user avatar
2 votes
0 answers
27 views

finite difference estimator of the derivative of the inverse distribution function

let $x_1,...,x_n$, $n\geq 3$ be a sample form the distribution $F$. We want to estimate $\frac{dF^{-1}(p)}{dp}$, replacing the distribution function $F$ by empirical distribution function $F_n$, and ...
Unknown's user avatar
  • 3,065
0 votes
0 answers
19 views

Expasion of Second Order Difference

I'm currently exploring a problem related to a two-parameter estimator, and I've simplified it for ease of discussion, eliminating the need for an extensive probability background. The problem is as ...
Frank's user avatar
  • 1
0 votes
0 answers
33 views

How does local and global error relate for finite difference methods?

When solving a (one-dimensional linear) boundary value problem by finite differences, how does the local and global error relate? I would have expected that if the local error was $O(h^k)$ in the step ...
ummg's user avatar
  • 561
2 votes
1 answer
217 views

Understanding a formula for the solution to an inhomogeneous linear difference equation

My question comes from section 8.4 of Analysis of Numerical Methods by Isaacson and Keller. I am trying to get a better understanding of the formula stated in Theorem 3, p.409. First, some definitions ...
Leonidas's user avatar
  • 908
0 votes
0 answers
32 views

Ensuring Symmetry in Mixed Derivatives Using RBF-FD Method

Hello Mathematics Stack Exchange Community, I'm working on a numerical problem where I have the first-order partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ of a ...
Rule's user avatar
  • 1
0 votes
1 answer
60 views

How best to generalize finite difference Laplacian matrix from 1D to 2D (and beyond)

To numerically solve the Poisson equation in one dimension we might transform the problem into a linear system $A u = b$, where $A$ is a finite difference approximation of the Laplace operator, say $$ ...
ummg's user avatar
  • 561
2 votes
0 answers
61 views

Did I get Godunov's scheme right?

I want to implement Godunov's scheme in order to simulate the nonlinear LWR-type equation $$ \partial_t u + \partial_x (u(1-u)) = 0, \quad u(0, \cdot) = u_0. $$ The update step is ($n$ denotes time ...
Hyperbolic PDE friend's user avatar
0 votes
0 answers
61 views

Boundary Value Problems Solving with Central Difference

I have differential equation $u''(x)+9u(x) = \cos(2x), \; x \in [0, π/2]$ the boundary conditions are $u(0)=1$, $u(\pi / 2) = -1$, with $h = 0.2$ Now I replaced $u''(x)$ with centered difference ...
lovbal's user avatar
  • 1
0 votes
0 answers
34 views

Curve Fitting and Parameter Estimation Study

I have a problem with an experimental data analysis to obtain unknown parameters. The experimental data can be described by the equation: $y_k-y_{k-1}=\eta_k (x_k^{1-\alpha_k}-x_{k-1}^{1-\alpha_k})$ ...
mcakir's user avatar
  • 1
1 vote
0 answers
13 views

Creating nonuniform grids for FDM with multiple points of concentration

If I am creating a grid in the $S_i$ direction with $N_S+1$ grid points. If I want more steps around some $K$, I can use: $$ S_i=K+c \sinh \left(\xi_i\right), \quad i=0,1, \ldots, N_S $$ where $c=\...
THAT'S MY QUANT MY QUANTITATIV's user avatar
0 votes
0 answers
47 views

Numerical Partial Differential Equation

The solution of the difference diagram for some partial differential equations from the Fourier transform and Fourier analysis can be written in ${U_{n}}^{k}=q^{k}e^{in\xi}$ form. The condition for ...
Yasemin's user avatar
1 vote
0 answers
175 views

Relationship Between Shift Operator and Backward Difference Operator (Finite Differences)

Consider $z_1,...,z_n$ and let $E(z_k) = z_{k+1}$ be a forward shift operator and let $\Delta_-(z_k) = z_k - z_{k-1}$ be a backward difference operator. I'm trying to show that $E = \sum_{j=0}^{\infty}...
Nicholas Gillespie's user avatar
0 votes
0 answers
28 views

Numerical Treatment of Dirac Delta in Finite Difference Scheme

I'm considering a particle in a spatial domain $\Omega \subset \mathbb{R}^2$ which secretes some chemical substance with production rate $q_0$, and is subject to a force proportional to the gradient ...
zaccandels's user avatar
0 votes
0 answers
29 views

Eigenvalue problem with boundary condition

I am playing with the one dimensional massless Dirac equation $$-i\partial_x \sigma_x \Psi(x) = \epsilon \Psi(x) \, .$$ Solving it analytically with some boundary conditions is an easy task. I am more ...
ibroketheinternet's user avatar
0 votes
0 answers
28 views

How to derive this finite difference scheme?

My professor gave me the following discretization scheme to use to discretize the mass conservation equation $\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$. I haven't seen this ...
aham_brahmasmi's user avatar
0 votes
0 answers
26 views

Finite Difference method, ADI Scheme of Douglas and Rachford

I am trying to implement the ADI scheme of Douglas and Rachford. For $p(X,Z,t)$, there is: $$ \begin{gathered} A=p^{n-1}+\Delta t_n\left[F_0\left(p^{n-1}, t_{n-1}\right)+F_1\left(p^{n-1}, t_{n-1}\...
THAT'S MY QUANT MY QUANTITATIV's user avatar
0 votes
0 answers
49 views

Finding second-order finite difference stencil

If you have the anisotropic diffusion equation to find $u(x,y)$ $$\left(\mu_{x}\frac{\partial^{2}u}{\partial x^{2}}+\mu_{y}\frac{\partial^{2}u}{\partial y^{2}}\right)=f(x,y) \tag 1$$ and you ...
blov's user avatar
  • 13
2 votes
0 answers
53 views

Can't get proper numerical convergence for complicated Advection-Diffusion-Reaction PDE

I have trying for quite some time to write a finite difference solver for the following advection-diffusion-reaction differential equation: $$ \frac{\partial C}{\partial x} = \frac{1}{u(z) + u_e(x,z)}\...
David G.'s user avatar
  • 260
0 votes
0 answers
18 views

Restructuring a Discrete Equation System for Compact Representation

I have the following system of equations for $i,j=1,...,n$: $u_{i-1,j}+u_{i+1,j}+u_{i,j-1}+u_{i,j+1}-4u_{i,j} = f_{i,j}$ The right hand side $f$ is known, as well as $u_{0,j},u_{n+1,j},u_{i,0},u_{i,n+...
Laneix's user avatar
  • 11
0 votes
0 answers
41 views

The real CFL condition for cylindrical laplacian

I've been exploring the CFL (Courant-Friedrichs-Lewy) condition in polar coordinates and have observed that previous inquiries haven't yielded a satisfactory answer. I've come across this paper which ...
Manuel Borra's user avatar
2 votes
1 answer
49 views

Transform Lagrangian with a square root

I have an action given by, \begin{equation} S = \int^{\tau_f}_{\tau_0} d\tau \left(\frac{1}{z^3}\sqrt{-f(u,z) \dot{u}^2 + 2 \dot{u} \dot{z} + \dot{x}^2} + \frac{2 \dot{z}}{z^3} \right), \end{equation} ...
mathemania's user avatar
0 votes
0 answers
40 views

Coupled non-linear PDE analytical solution to verify numerical solution.

as part of a computational module we were given the following coupled PDEs and solved them numerically using finite difference methods: I got the following graphs representing the numerical solution ...
Nicojwn's user avatar
  • 103
2 votes
1 answer
45 views

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}{...
David G.'s user avatar
  • 260
0 votes
0 answers
30 views

second-order difference equation (hypergeometric type equation)

We defined the backward and forward difference operators ($\Delta$ and $\nabla$ respectively) by $\Delta f(x)=f(x+1)-f(x)$ and $\nabla f(x)=f(x)-f(x-1)$. We consider the following equation $$ -B(x)f(x+...
Made's user avatar
  • 1,249
0 votes
0 answers
43 views

Finite Difference Method for Poisson's Equation

I'm writing a Python program to solve Poisson's equation $$ \nabla^2 u = f \quad \mathrm{on} \quad \Omega $$ with $$ \frac{\partial u}{\partial n} = 0 \quad \mathrm{on} \quad \partial \Omega. $$ Here,...
zaccandels's user avatar
7 votes
1 answer
174 views

If $f((\text{deg}(f) + 1$) consecutive $\mathbb{Z}\text{s}) \subset \mathbb{Z}$ then $f(\mathbb{Z}) \subseteq \mathbb{Z}$

Question: Let $f:\mathbb{R}\to\mathbb{R}$ be a polynomial of degree $n$. Suppose there exists an integer $m\in\mathbb{Z}$ such that $f(m)\in\mathbb{Z}$, $f(m+1)\in\mathbb{Z}$, ..., $f(m+n)\in\mathbb{...
IraeVid's user avatar
  • 3,161
1 vote
0 answers
42 views

Discretize Burger's equation with upwind strategy

Let the 1D burgers equation be defined by $\frac{\partial u}{\partial t} + a \frac{\partial u}{\partial x} = v \frac{\partial^2 u}{\partial x^2}$ where $v = \frac{1}{Re}$ where $Re$ is the number of ...
dabib's user avatar
  • 11
0 votes
0 answers
19 views

How to handle the evaluation of functions on staggered ghost nodes?

I have a convection-diffusion-reaction steady state PDE in the form $$ u(x,z)\frac{\partial C}{\partial x} = \frac{\partial}{\partial z} \left( K(z) \frac{\partial C}{\partial z} \right) - \lambda C - ...
David G.'s user avatar
  • 260
0 votes
1 answer
62 views

Why does my finite difference implementation for a 1-D 4th order differential equation not work?

I am trying to solve a 1-D 4th order differential equation using finte difference method. I am implementing the algorithm in maple and was able to get approximate solutions for second order ...
MazzMan's user avatar
  • 115
0 votes
0 answers
28 views

Is this the correct way to map finite differences from the radial to the exponential coordinate system?

I am currently numerically solving the following partial differential equation using finite differences: $$ C {\partial ^2v \over \partial t^2} = {\partial ^2v \over \partial r^2} + { 1\over r}{\...
Nikola Ristic's user avatar
0 votes
0 answers
36 views

Three term recurrence relation with two indices and variable coefficients

Let $c \in \mathbb{R}$. I would like to solve the following non-constant, two-indices recurrence relation on $u = (u_j^n)^{n=1, 2, \dots}_{j=0, 1, \dots}$: $$ u_j^0 = 0, \qquad \text{for} \quad j \geq ...
94thomas's user avatar
0 votes
0 answers
17 views

Need help setting up a staggered grid. I have never used a non-collocated grid before.

From what I've read so far, there are the scalar variables and vector variables that make up finite difference equations that would all normally be placed on a coordinate point (i,j) on a collocated ...
Researcher R's user avatar
0 votes
1 answer
54 views

Generating uniform grids

I have the following first order differential equation: $$0.05u'+2xu=0.05^2e^{5-20x^2}$$ for $x \in [0,1]$, and $u(0)=0$. then I want to approach the differential equation with a Taylor series at ...
RandomlyX's user avatar
2 votes
1 answer
88 views

Lax-Friedrichs Scheme for second-order hyperbolic PDE

For the second-order hyperbolic PDE on $\Omega = (0,1)$: $$ u_{tt}(x,t) = c^2 u_{xx}(x,t)\; , \quad \begin{cases} u(0,t) = L(t) \\ u(1,t) = R(t) \\ u(x,0) = f(x) \\ u_t(x,0) = g(x) \...
Mak's user avatar
  • 21
0 votes
0 answers
57 views

Lagrange multiplier method in a discretized way

I succeeded in using the Lagrange multiplier method to solve continuous case, but I failed to solve discretized case. Assume $\min f(x,y) = x^2 + y^2$ constraints $s.t. g(x,y) = x + y -1 = 0$ use ...
sdfa's user avatar
  • 1

1
2 3 4 5
17