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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19 views

Grid Independent Study

What is the difference Between local time stepping and adaptive time stepping? Is the change in time step necessary for the grid independent study? As there is relation between dt and dx in CFL. In ...
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1answer
59 views

Imposing no flux boundary conditions on variants of the Cahn-Hilliard equation using Finite Differences in Python

I have been looking into simulations of phase separation in variants of the Cahn-Hilliard system and have been running into issues with implementing no flux boundary conditions on certain variants. ...
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164 views

Solve wave equation with discontinuous coefficients numerically?

I would like to solve the following equation $$\frac{\partial^2 y}{\partial t^2} - c^2(x,t)\frac{\partial^2 y}{\partial x^2}=0,$$ for $y=y(x,t)$ numerically. The wave speed, $c(x,t)$, is of the form $$...
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21 views

Choosing appropriate timestep for FDTD and PIC algorithms

I'm doing some particle-in-cell (PIC) simulations of propagation of short pulses in plasmas with a widely used code. This code uses Yee's field solver. So, the timestep must be lower than the Courant ...
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1answer
324 views

Suitable finite difference method for a convection-diffusion system?

I am trying to solve a system of PDEs $H_{t} = \frac{0.3}{0.7} - \frac{0.005 B f(h(H))}{\theta} - \frac{0.3 f(h(H))}{0.7} + \frac{500}{0.7} (HH_x)_x + (HH_y)_y$ $N_t = \frac{N_{in} - 0.002 [N] B f(h(...
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343 views

When and when not to use automatic differentiation

I am just learning (more) about automatic differentiation (AD) and at this stage it kind of seems like black magic to me. The second paragraph of its Wikipedia article makes it sound too good to be ...
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53 views

Preconditioning the $[1 \quad-2 \quad 1]$ Finite Difference matrix

Let $A$ be the well known tridiagonal matrix coming from the 1D Finite difference discretization of the Laplacian, with stencil $\frac{[1 \quad-2 \quad 1]}{h^2}$. The system $Ax = b$ is very large, so ...
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146 views

$P0$ elements for $H1$

Are there $P0$ (zero degree/constant element) nonconforming methods for approximating solutions in $H1$? More specifically, I have the equation: $$u-f - T\Delta u = 0$$ Which can be interpreted as ...
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1answer
38 views

How to calculate error in successive over relaxation for PDE?

I am trying to solve the Poisson equation numerically using the FDM method in C++. But I have a little confusion with the iterative process. I understand that the number of iterations should go until ...
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1answer
95 views

Parabolic differential equations with time delay

Let $d_1=1,d_2=2,a_{11}=\frac{5}{13},a_{12}=\frac{22}3,a_{21}=-2,a_{22}=\frac{6}7,\tau=\frac{5}7$, $\psi(t,x)=\cos^42x,\phi(t,x)=\frac{3}{13}x^4\sin^2 3x$, $\Omega=[0,200]$ How to solve: $$\left\{ \...
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82 views

Which scheme for inhomogeneous convection-diffusion equation with highly variable coefficients?

I have a 1D convection-diffusion equation $\sigma_t = a(x,t) \sigma_{xx}+b(x,t)\sigma_x+f(x,t)$ defined on the unit interval, with nonzero Neumann boundary conditions at both ends. It should be noted ...
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1answer
60 views

Effect on methods like Crank-Nicolson of adding a potential term, changing heat equation to Schrodinger equation

I'm studying up on methods for numerically solving the Schrodinger equation. The Schrodinger equation with a zero potential is formally identical to the heat equation in the sense that we just make ...
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1answer
177 views

Solving nonlinear PDE with finite difference based on Newton-Krylov

I am now working on solving MHD equations with finite difference method, which include nonlinear equations: $$ \frac{\partial\rho}{\partial t}+\nabla\cdot\left[\left(\rho_0+\rho\right){v}\right]-\...
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7k views

Use of machine learning in computational fluid dynamics

Background: I have only built one working numeric solution to 2d Navier-Stokes, for a course. It was a solution for lid-driven cavity flow. The course, however, discussed a number of schemas for ...
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1answer
180 views

Is there a simple way to avoid carbuncles for FD WENO methods?

I have implemented finite-difference WENO scheme for Euler equations (with some variants - WENO-JS, WENO-Z, WENO-M, different flux splitting). It works well, but have problem with so-called carbuncles ...
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2answers
98 views

Finite difference method having a discontinuity

I am trying to understand the FDM which is a widely used method solving differential equations by using approximation below. $$\dfrac{\partial u}{\partial x}=\dfrac{u(i+1)-u(i-1)}{2\Delta x}$$ How can ...
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Numerical calculation of the Berry connection

I'm doing some numerical calculations involving Hermitian matrices, and derivatives of the eigenvectors. Essentially, I have an n x n, Hermitian matrix H(x), which is dependent on some continuous ...
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1answer
46 views

2D heat equation in both steady state and Transient state using iterative solvers

While solving a 2D heat equation in both steady-state and Transient state using iterative solvers like Jacobi, Gauss seidel, SOR. Should the answers, I mean the converged results of Temperature ...
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1answer
81 views

Finite difference methods in cylindrical and spherical co-ordinate systems

I am quite familiar with finite difference schemes in cartesian coordinates. The key point here is that every point in the cartesian grid is treated equally as the spacing between consecutive points ...
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1answer
30 views

How to fill matrix entries for two-dimensional implicit finite-difference for the general case

If I have derived a finite-difference formula for a 2D problem, for example something like: $af_{i,j}+bf_{i-1,j}+cf_{i,j-1}+df_{i-1,j-1}=g_{i,j}$ where f is the unknown function on a grid and ...
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106 views

Kronecker product representation of the finite difference laplacian

The laplacian equation when discretized gives a system of linear equations that can then be solved. See the answer to this question: https://math.stackexchange.com/questions/3120948/discretization-...
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1answer
90 views

Numerical methods that can be written in flux conservative form

I have a system of non-linear PDEs that I expect to have shocks as well as the appearance of Gibbs phenomena (spurious oscillations that form near the shock) for 2nd-order methods or higher. I have ...
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45 views

When to discretize nonlinear Poisson Equation

I am trying to solve a nonlinear poisson equation of the form: $u_{xx} + f(u_y)u_{yy} = 0$. In trying to get a handle on this problem, it seems like there are two approaches. I could either ...
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1answer
108 views

Order of accuracy for finite-difference on nonuniform grid

If we evaluate the first derivative of a function F(x) on a 1D grid {$x_i$} by central difference at $x=x_i$ as $$ \frac{dF}{dx} \approx \frac{F_{i+1} - F_{i-1}}{x_{i+1} -x_{i-1}} $$ then it is ...
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1answer
69 views

Finite Difference libray C++

What is the best FD library (or collection of libraries) for C++ codes? I am looking for some data structure implementation that offers the possibility to do parallel computations on adaptively ...
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1answer
61 views

FDM on nonlinear PDEs

I'm working with a 2D Navier Stokes PDE in the unstabilized version - the equation is a linear equation of the type $\frac{∂u}{∂t} = F(u,t)$. In order to perform time discretization with FDM (finite ...
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8answers
5k views

What programming language should I choose and why?

I am a mechanical engineer, intermediated/advanced level in MATLAB and MATHEMATICA, and beginner in Python. I intend to get a PhD in aeroelasticity (FEM + CFD) and coding my own program. I intend to ...
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20 views

Convection equation expanded in Legendre polynomials

In some areas of physics, for example radiation transport in plasmas, it is beneficial to re-express the convection equation in terms of a Legendre polynomial expansion. This results in a set of ...
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1answer
28 views

Norm of operator in finite element discretization of Heat equation

I am solving the heat equation discretized spatially via FEM and temporally via backward Euler. I get the system $$M \dot{u} = K u +f$$ where $u$ is a vector representing the solution at spatial ...
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38 views

How does the “Stable Fluids” algorithm by Jos Stam relate to the SIMPLE and PISO algorithms?

The "Stable Fluids" paper (*) by Jos Stam starts by acknowledging that "Our method cannot be found in the computational fluids literature, since it is custom made for computer graphics applications. ...
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How to reorder variables to produce a banded matrix of minimum bandwidth?

I'm trying to solve a 2D Poisson equation by finite differences. In the process, I obtain a sparse matrix with only $5$ variables in each equation. For example, if the variables were $U$, then the ...
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1answer
83 views

Efficient Arbitrary Order Finite Differences in 1D

I am implementing on Matlab a high-order finite differences scheme to approximate the first derivative of $f(x_i)$ given $x = [x(1), x(2),..., x(i),..., x(n)]$ and $f = [f(x(1)),..,f(x(n))]$ with $x$ ...
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26 views

von Neumann analysis: computation of maximum value of amplification factor

In this question I adress the stability property of a numerical scheme (a FDM described below) to the problem $$ \begin{align} u_{t} &= \alpha u_{xx}+\beta u_{xxxx}, & &x\in\mathbb{R},\;t&...
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3answers
155 views

How is central difference scheme second-order accurate?

In an arbitrarily unstructured mesh, shown in the figure below, in the context of finite volume method, I want to obtain an approximation of $\phi_f$, where $N$ and $P$ are cell centers of adjacent ...
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1answer
55 views

Solving the pulse propagation using four different FDTD methods gives four different results - Which to trust?

I'd like to simulate the propagation of a pulse, and have different options for solving that. On the one hand, I can use the non-linear schrödinger equation $$\partial_zE=\frac{i}{2k_0}\nabla^2_\perp ...
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339 views

How to make a less diffusive code to solve 2D advection equation?

I would like to solve the following differential equation numerically in 2D, $$\frac{\partial z^-}{\partial t}+(\vec{B}\cdot\vec{\nabla})z^-=0,$$ see Wikipedia if you are curious about what the ...
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How to determine the finite difference coefficient matrix in 2D with periodic BC?

I'm solving a PDE in matlab using ode15s, and since the spatial dimension is 2, and number of variables grow large very quickly, I need to supply the structure of ...
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1answer
74 views

Crank-Nicholson scheme for transport equation

This is my attempt to find the approximate solution of the folowing transport equation $$\left\{\begin{array}{ll} \partial_{t} u+\partial_{x} u= (x^2-x)t+x^3/3-x^2/2, & t \in(0,0.4), x \in(0,1) \\ ...
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1answer
109 views

Time complexity of numerical finite differences

I have a function $f:\mathbb R^N\to \mathbb R$ and I would like to compute all the partial derivatives of $f$ w.r.t. the $N$ input. What is the computational complexity using the (ones-sided) finite ...
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52 views

What space-time points should a known coefficient function be evaluated at when using the Lax-Friedrichs scheme to solve the transport equation?

For a scalar quantity $u = u(x, t)$, I'm considering the transport equation \begin{align} u_t + au_x &= 0, \qquad{x\in[0, L], \ t\in[0, T]} \\ u(0, t) &= u_{\text{in}}, \\ u(x, 0) &= f(x). ...
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Spectral solver on em-pic

I'm recently studying for the spectral solver to implement EM-PIC code. I read an article and have some questions. Many PIC codes uses spectral solver to overcome numerical artifacts on FDTD. In the ...
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1answer
78 views

What is the maximum attainable accuracy with a given set of $\alpha,\beta$?

I am using LeVeque's book: https://faculty.washington.edu/rjl/fdmbook/ Suppose I want to compute $u''$ using FDM with $\alpha=\beta=2$ (centered) so the FDM is $$ u''=\sum_{m = - \alpha}^\beta a_mu(...
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Convergent Finite Difference Scheme for Parabolic Equation

Consider the PDE $$u_t = b_{11}u_{xx} + 2b_{12}u_{xy} + b_{22}u_{yy},$$ where $b_{11}, b_{22} > 0$, and $b_{12}^2 < b_1b_2$. In Strikwerda's book, the ADI scehme \begin{align*} \left( 1 - \frac{...
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1answer
101 views

Simple Harmonic Motion using the leapfrog method

I have to use the leapfrog method to solve the simple harmonic oscillator and I having trouble writing it in code. This is what we were given in class $$ \frac{v_{n+1/2}-v_{n-1/2}}{\Delta t}=-\omega_0^...
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6answers
15k views

How can I numerically differentiate an unevenly sampled function?

Standard finite difference formulas are usable to numerically compute a derivative under the expectation that you have function values $f(x_k)$ at evenly spaced points, so that $h \equiv x_{k+1} - x_k$...
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Numerical errors due to terms of the form $\frac{1}{r}$ (r goes to 0 at the boundary) while using finite difference method

I am trying to solve a system of differential equations using finite difference method. There are few terms of the form $\frac{A(r)}{r}$, both $A(r)$ and r go to zero at the boundary. Analytically ...
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63 views

Are linear, CTCS codes always stable?

I would like to solve some equations which basically look like this $$\frac{\partial u}{\partial x}=F\left(v,\frac{\partial v}{\partial y},\frac{\partial^2 v}{\partial y^2}\right),$$ $$\frac{\partial ...
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1answer
59 views

Solve convection-diffusion equation with a non-linear source term

I would like to solve this equation (which is adapted in my case, a plug flow reactor when a reaction occurs): $ \frac{df}{dt} = D \frac{d²f}{dz²} - u \frac{df}{dz} + r(z,t) $ with $r(z,t)= - k f^{n}...
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1answer
190 views

Order of Accuracy Measurements on 1D Advection Methods

I am trying to learn about basics of computational fluid dynamics, at the moment on the simple example of linear advection in 1D. I am am currently testing the theoretical predictions of the order of ...
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1answer
30 views

Truncation error plot with weird issue

I have a function f(x) = sin(x)/x^3 whose first derivative I am trying to estimate using 1st, 2nd and 4th order Finite Difference schemes. I tried to plot the ...

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