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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1answer
67 views

Accuracy of Explicit Euler method (finite difference) decreases as Δx decreases, shouldn't it increase?

The price of a commodity can be described by the Schwartz mean reverting SDE $$dS = \alpha(\mu-\log S)Sdt + \sigma S dW, \qquad \begin{array}.W = \text{ Standard Brownian motion} \\ \alpha = \text{ ...
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1answer
49 views

How is the mixed 2nd partial derivative simplified to a more efficient form?

I'm implementing the Finite Different 2nd and 3rd derivatives in my research and naturally I'm looking for the most efficient approach. From https://en.wikipedia.org/wiki/Finite_difference#cite_ref-...
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1answer
67 views

If analytically, a function is not differentiable at a point, does it make sense to write a finite difference code for the function at that point?

What would happen if I wrote a finite-difference code evaluated at a point where the function isn't differentiable analytically? I'm trying to think analytically vs numerically. Thanks,
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93 views

Solution of Cahn-Hilliard equation

I need to solve the Cahn-Hilliard equation $$\frac{\partial u}{\partial t} = \Delta(f(u) - \epsilon^2\Delta u), \hspace{.5cm}(x, t)\in \Omega\times(0, T],$$ using mixed formulation \begin{equation}\...
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1answer
60 views

3D laplacian operator

I have been unable to find the equivalent of the 5-point stencil finite differences for the Laplacian operator. In 2 dimensions for me it is clear that, using the finite difference method: $$ \nabla_{...
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1answer
57 views

Parity for artificial dissipation term in a finite-difference solution

I have a doubt regarding the signal of the dissipative term in a finite difference solution for an equation of the form $$ \frac{\partial u}{\partial x}+f(x)=0, u(0)=0 $$ In which $f$ is an odd ...
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41 views

numerical solution to pde on an ellipse

Looking for advice on discretization (preferably finite difference) schemes for pdes on curves in general, but in this case it is an ellipse (so given by $(a\cos(r), b\sin(r)$). The problem is the ...
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2answers
50 views

No flux Neumann boundary condition for non-stationary PDE equivalent to Dirichlet boundary?

When using no flux Neumann boundary conditions (i.e. zero derivative to the normal on the boundary) in a non-stationary PDE, I don't seem to recognize the difference to using Dirichlet boundary ...
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1answer
71 views

Method of Lines: How to simplify Jacobian with periodic BCs?

Consider the advection equation $$\frac{\partial u}{\partial t}+c(x)\frac{\partial u}{\partial x}=0.$$ With periodic boundary condtitions in $x$ with period $L$, i.e. $u(x,t)=u(x+L,t)$ and initial ...
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3answers
316 views

Finite-difference software for solving custom equations

Are there any good, easy to use, software for simulating the evolution of systems of generic differential equations? I know there are custom programs for various specific circumstances (such as ...
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1answer
89 views

Interpolating the gradient of a cylindrically symmetric potential field that's 'supposed to' obey the Laplace equation?

The script below tries to implement a Jacobi iterative relaxation of a potential field for an electrostatic lens. It's hot-off-the-press and I've just started to debug and look for things to test it ...
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1answer
54 views

Problems with manufactured solutions for 1D inviscid burgers' equation

I'm having an issue with the easiest example of a nonlinear 1D PDE, the (inviscid) burgers' equation: $u_t + uu_x = 0,~~(1)$ which can be rewritten as some convection equation $u_t + f(u)_x = 0$ with ...
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1answer
171 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 known ...
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1answer
373 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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0answers
54 views

1-D Conduction Steady state heat transfer using FD

I have tried to solve 1-D Conduction Steady state heat transfer problems in Matlab (see below). Here is the 1-D model: T''[x] == 0, T[0] == 100, T'[100] == 0 ...
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1answer
87 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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2answers
187 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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0answers
25 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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1answer
357 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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54 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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2answers
149 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
39 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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1answer
84 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
61 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
183 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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2answers
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 ...
2
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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
111 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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0answers
40 views

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
49 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 ...
3
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1answer
90 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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2answers
117 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
95 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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0answers
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
79 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
65 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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0answers
21 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
31 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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0answers
41 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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6answers
2k views

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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0answers
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
160 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
56 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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0answers
42 views

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
76 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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