Questions tagged [finite-difference]

Referring to the discretization of derivatives by Finite differences, and its applications to numerical solutions of partial differential equations.

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Finite difference problem

I have a problem to resolve with the Finite Difference method in $[a,b]$: $$-\frac{d}{dx}(\alpha(x)\frac{du}{dx})= g(x),$$ with $\alpha(x) \in L^{\infty}$ continuous in $]a,c[$ and $]c,b[$ and ...
Kaneki Ken's user avatar
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Is a sort of "z-drift" the result of numerical precision errors in FDM?

Upon solving the 2D wave equation with Neumann boundary conditions $u_x = u_y = 0$ on a rectangular $10 \times 10 \times 10$ grid, I noticed something odd - $u$ seemed to shift upwards with time. This ...
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Stability of Euler forward method

I am trying to solve a linear system of ODEs of the form: $$ \frac{du}{dt} = A u, \quad u(0)=k$$ where $A$ is a 2x2 matrix and $u(t)$ is a 2x1 column vector. I want to solve this numerically, using ...
rainbow's user avatar
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derivative matrix and the Dirac delta distribution

For a project I'm working on, I was working with the following equation $$ w(x) = \int k(x,y)v(y)dy $$ I noticed that if I choose $$ k(x,y) = -\delta'(x-y) $$ Then we probably get (I haven't touched ...
NNN's user avatar
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Does anyone know how to add a forcing term at the center of a cicular membrane?

I am here once again searching for wisdom, as some of you might notice I asked a related question a few days ago. And now I am struggling to add a forcing term at the center of the membrane, in order ...
Manuel Borra's user avatar
2 votes
1 answer
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Can this finite difference dispersion be eliminated somehow?

I am trying to solve the wave equation $$ {\partial ^2u(t,x) \over \partial x^2} = {\partial ^2u(t,x) \over \partial t^2} \tag1 $$ with the following boundary and initial conditions: $$ {\partial u \...
Nikola Ristic's user avatar
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Solving the wave equation for a circular membrane in polar cordinates

As you see this mode is not right, unless for what i understand And the initial conditions were ...
Manuel Borra's user avatar
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First-order modified Patankar–Euler scheme (MPE)

Is the first-order Modified Patankar–Euler scheme (MPE) an implicit or explicit method? Is there an open-source code implementing the MPE scheme for a system of ODEs?
Mahmoud Saleh's user avatar
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Discretization of generalized kinetic term in 2D Poisson partial differential equation

A typical 2D Poisson PDE is given as $$\nabla^2\varphi(x, y)=f(x, y)$$ where the Laplacian term, $\nabla^2\varphi$, can to some degree be interpreted as the kinetic energy (given proper scaling) (...
Akhaim's user avatar
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Perfect Conducting (PC) and Perfect Insulating (PI) boundary conditions for magnetic induction in MHD problems in cylindrical coordinates

I understand the computational implementation of the PC and PI bcs in cartesian case. In terms of the magnetic field components for PI, we have $B_x = B_y = B_z = 0$, and for PC, we have $\partial_x ...
myresh's user avatar
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Modeling contamination diffusion in a draining container, part 2

Part 1, but I'll repeat here. This time we'll move the top boundary. I have a container of water with initial volume $V_0$ and height $L$, open at the top which receives a constant mass flux $\phi$. A ...
HiddenBabel's user avatar
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Closed formula to diagonalize discretized (perhaps randomized) Laplacians

I was wondering whether there is a closed formula for the eigenvalues and eigenvectors of the discretized Laplacian in (edit) $[0,1]^n$ with a uniform grid, using what I imagine is a $2n+1$ stencil. ...
Aner's user avatar
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Automatic Differentiation In the Presence of Jump Points

I have a complex monte-carlo cashflow model that traditionally uses the finite difference (FD) method to calculate its derivative at any given point. To improve model performance, I coded forward-mode ...
Mild_Thornberry's user avatar
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Modeling contamination diffusion in a draining container

I asked this on the Physics site, but it fits much better here, and I now can ask a more specific question. For the scope of the problem, I have a 250-mL bottle filled with pure water and a sampling ...
HiddenBabel's user avatar
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Can any discretization scheme reproduce the kane quasi-linear dispersion relation?

It is straightforward to solve the eigenproblem $$\hat{k}^2 \psi = E\psi$$using a finite-difference method. But what about$$\hat{k}^2 \psi = E(1+\alpha E)\psi$$I know it is possible to first solve the ...
DJames's user avatar
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First derivative central differences with reflecting boundary conditions

I have the following problem in 1-D: \begin{equation} \partial_t u(t,x) = v(x)\partial_x u(t,x) + \kappa(x) \partial_{xx} u(t,x),\,x\in\Omega;\quad \partial_{\nu}u(t,x) = 0, \,x\in\partial\Omega. \end{...
lightxbulb's user avatar
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Stability for the 2d diffusion equation

I'm reading this set of notes on numerical stability of the diffusion equation. I can't seem to derive equation (7.17) from the previous equation. I've tried using $\sin^2(\theta)=\frac{1-\cos(2\theta)...
NNN's user avatar
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2 answers
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How and when to impose inhomogeneous Dirichlet boundary conditions in Newton method solver for a PDE?

I am applying a central Finite Difference scheme in space and an implicit Euler scheme in time on a variant of the 2d Burgers equation, of the form:$$u_t + uu_x + uu_y = \nu(u_{xx} + u_{yy})$$ where $...
Robby Ram's user avatar
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Literature request for pinning the corner singularities in finite differences

Corner singularities cause instability of finite difference solutions. They occur at the intersection of boundary conditions. I was recently going through an online video course that was explaining ...
Nikola Ristic's user avatar
3 votes
1 answer
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Solving Poisson's Equation with Periodic Boundary Conditions

So, I've been attempting to design a simple solver for a problem of finding the gravitational potential of a system using Poisson's equation (let's call the potential phi, $\phi$). The goal is that I ...
TheAkashain's user avatar
2 votes
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108 views

Efficient heat diffusion implementation with varying coefficients

I have the following heat diffusion equation: \begin{alignat}{3} \partial_t u(t, \vec{x}) &= g(\vec{x})\Delta u(t,\vec{x}), &\quad& \vec{x} \in\Omega, \, t\in(0,\infty],\\ \partial_n u(t,\...
lightxbulb's user avatar
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Generalized eigenvalue problem for large, potentially ill-conditioned systems

Say that I have a generalized eigenvalue problem of the form $$Ax=\lambda Bx.$$ Using MATLAB, some naive ways that one may solve this is by either directly inverting $B$ then applying the ...
user45844's user avatar
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Can I apply the product rule for the following finite difference discretization

I would like to know if the following discretization is correct. Here D is the dispersion, C is the concentration. Both D and C are varying with space. Here n+1 represents the unknown time level. I ...
Jaywalker's user avatar
1 vote
1 answer
102 views

Isolating decaying solutions to nonlinear second-order ode

I need to solve a nonlinear ODE of the form $$ \frac{d^2 y}{dx^2} + \frac{1}{x}\frac{dy}{dx}-\frac{1}{x^2}\sin(y)\cos(y)+\frac{2}{\alpha}\frac{\sin^2(y)}{x}-\sin(y)=0 $$ numerically, subject to the ...
Ali Shakir's user avatar
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28 views

Parallel Block-Structured class abstraction for FDM

I’m currently developing a FDM/FVM (using contravariant coordinates) code using Fortran and Co-Arrays (SIMD, in general), and so far I have all sparse matrix (BiCGStab, working on AMG) solvers and ...
Kbzon's user avatar
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How can i model upward natural convection in various angles (0 - 90)?

I am looking for a way to numerically solve the Naiver-Stokes equations for steady incompressible flow using FDM over a surface with various angles?
user16829029's user avatar
1 vote
1 answer
144 views

Crank Nicolson Method with closed boundary conditions

I want to simulate 1D diffusion with a constant diffusion coefficient using the Crank-Nicolson method. $$\frac{\partial u (x,t)}{\partial t} = D \frac{\partial^2 u(x,t)}{\partial x^2}.$$ I take an ...
Jbag1212's user avatar
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Oscillation in non-linear porous flow solved by finite difference

I am trying to solve numerically the flow of a gas through a porous, spherically symmetric body. The non-dimensional equations read: $$ \frac{\partial\rho}{\partial t}+\tau\frac{1}{r^2}\frac{\partial}{...
MaximeMaurice's user avatar
6 votes
1 answer
129 views

Numerical artefacts in solution of spherical heat equation using FDM

I was attempting to solve the diffusion equation for a solid sphere using a naive FDM scheme. The governing PDE for the scalar concentration field $u(r,t)$ is $$ u_t = r^{-2}(r^2 \alpha u_r)_r, \quad ...
IPribec's user avatar
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fdtd ansys Lumerical vs fresnel solution

I am comparing reflectance intensity of silicon structure with a hole. I am getting oscillating solution for 0 roughness factor in equation based model. but spectrum shape of fdtd and equation based ...
aawer's user avatar
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11 votes
1 answer
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Is using iterative methods to solve a linear system always superior to inversing the matrix?

I have a silly question. Is it always more computationally efficient to use iterative methods to solve for some matrix $A$, $Ax=b$, where $x$ and $b$ change but $A$ stays constant, compared to ...
Touko Puro's user avatar
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1 answer
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How to address the element face adjacent to boundaries when the finite difference method and marker-and-cell scheme are used to solve the Stokes flow?

The Stokes equations are $$-\Delta \mathbf u + \nabla p = f \text{, in }\Omega,$$ and $$ -\nabla \cdot \mathbf u = g, \text{ in } \Omega$$ where $\mathbf u = \left( u, v \right)$ is the flow ...
Tingchang Yin's user avatar
1 vote
1 answer
112 views

how to solve a coupled nonlinear partial differential equations(Boundary Value Problem)

So far, I have been looking into linear and nonlinear differential equations(Boundary value problem) that look like the following: $\frac{d^2 \theta}{dz^2} + \sin(\theta) = 0$ I am now familiar with ...
Hari Sam's user avatar
3 votes
0 answers
150 views

Python code of explicit method of a nonlinear a BVP

I am trying to have a Python code for the following nonlinear BVP: $$\frac{\partial N}{\partial t}=\frac{\partial^2 N}{\partial x^2}+N(1-N)-\sigma N$$ $$N(0,x)=\sin(2\pi x)$$ $$N(t,0)=0 \hspace{3mm}N(...
Peachy April's user avatar
1 vote
1 answer
61 views

Once Lyapunov exponents have converged, can they diverge again?

I know the strength of attraction for a single attractor might vary from place to place. Say, if I calculated the Lyapunov exponents for a small portion of the attractor and they have already ...
Axel Wang's user avatar
2 votes
1 answer
112 views

How to numerically solve differential equations involving sines, cosines and inverses of the unknown function?

I'm very new to finite difference method and I am just introduced to methods of solving differential equation using finite difference method via sparse matrix method. I find that the main idea is to ...
Hari Sam's user avatar
1 vote
3 answers
226 views

Partial derivatives for triangular meshes (in 3D)

A grid offers an obvious definition for the partial derivatives at a grid point, given $x$ the value of a point $p$ in an $n$ dimensional grid, the forward partial derivative that point for coordinate ...
Makogan's user avatar
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Refluxing step on Finite difference AMR

Hi I am a computer scientist working on MHD code for astrophysics simulation. We use a finite difference scheme where we first solve the spatial derivatives and with them solve the right hand side and ...
Touko Puro's user avatar
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0 answers
74 views

Encountering blow-up when solving the one-way heat equation using Lax-Wendroff

This is my first time attempting to implement a finite difference method for a PDE in Python, and I am having a bit of trouble. The PDE I am trying to solve is as follows: $$ \begin{cases} ...
Leonidas's user avatar
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Need help implementing finite difference Beam Propagation Method to Solve 2-D Helmholtz equation

I am trying to implement beam propagtion method in a two-dimensional lattice to solve Helmholtz equation by following the scheme given this paper. I am using Matlab for implementation. The expected ...
Yashab Yadav's user avatar
3 votes
1 answer
89 views

A question related with $p$-Laplacian and conjugate gradient method

I have the following energy functional of $p$-Laplacian equation: $$ E(u) = \frac{1}{p} \int_{\Omega} |\nabla u|^p dx $$ for $2.8 \leq p \leq 5$. My goal is to minimize the energy functional by using ...
User124356's user avatar
2 votes
0 answers
146 views

Poisson equation solution in a semiconductor structure

I am trying to solve the $\textbf{1-D}$ Poisson equation for a semiconductor structure at equilibrium (There is no external bias applied). $\textbf{Background}$ \begin{equation} \frac{d^2V}{dx^2} = -\...
Photon's user avatar
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1 answer
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On solving a first order nonlinear differential equation

It all starts with this Cauchy problem: $$ \begin{cases} \sin(2x(t)) -\cos(3x'(t)) = x(t) + x'(t) \\ x(0) = 1 \\ \end{cases} \quad \quad \text{with} \; t \in [0,10]\,. $$ Not knowing which way to turn,...
Monster's user avatar
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2 votes
2 answers
227 views

Solving 2D Poisson equation with nonhomogeneous boundary conditions (Dirichlet) and a source

I am a physicist who is fairly new to numerical analysis, currently, I am trying to simulate a non-linear paraxial equation, and part of my calculation involves solving a 2D Poisson equation with ...
Omer Paz's user avatar
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55 views

Solving Laplace for Velocity Potential in Constricted Channel

I am trying to solve the 2D Laplace equation numerically to give the velocity potential of a fluid flowing in a channel with a constriction: $u_{xx}+u_{yy}=0$ There is a constriction in the channel at ...
Not a Mathematician's user avatar
2 votes
0 answers
100 views

Scipy.root not converging even when provided with initial guesses very close to solution

I've made a previous question here and also in SO wondering why only the fsolve solver converges for the simple one dimensional unsteady conduction problem $$ \frac{\partial T}{\partial t} = \alpha \...
Klaus3's user avatar
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2 answers
305 views

Why is this scipy.root code not converging?

I'm running a test problem to set up larger problems. Solving the simple unsteady heat equation via finite differences: $$ \frac{\partial T}{\partial t} = \alpha \frac{\partial^2T}{\partial x^2}$$ $\...
Klaus3's user avatar
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1 vote
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How to include zero flux boundary conditions?

I am trying to solve the following differential equation in the domain of $\theta \in [0, 2 \pi]$ using finite differences scheme: For $0< \theta \leq \pi$ \begin{align} \rho_i^{n+1}=\rho_i^{n}+D\...
Irbin B.'s user avatar
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1 vote
0 answers
101 views

A staggered grid for an eigenvalue problem (linear stability analysis)

I'm interested in extending the concept of a staggered grid (commonly used to solve the incompressible Navier-Stokes equations) to a linear stability analysis context. For example, we can consider ...
Samantha B.'s user avatar
2 votes
1 answer
134 views

Computing numerical derivatives

I am trying to create a sweeping surface, for which I need the frenet frame of a curve. I am trying to compute this for arbitrary curves but for testing I am just using the parametric unit half circle....
Makogan's user avatar
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