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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does finite element perform better than finite difference on AMR meshes?

my current work requires using AMR to resolve multi scale physics. I have a general question whether finite element is better than finite difference in this aspect or not. i know that finite element ...
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32 views

Mineral dissolution and solute transport around a solid

I am trying to simulate solute transport of acid (HCl) and consequent mineral dissolution around a grain (calcite). The governing equation for transport is the advection-diffusion equation, given as: ...
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How to troubleshoot numerical instability using finite difference for steady-state non-linear heat conduction equation

I have a problem which I believe is numerical instability when trying to solve a heat conduction equation using finite difference. The short version is that when the parameter $I=80.3$ I get the blue ...
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1answer
398 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
71 views

1D wave equation using Finite difference method MATLAB

I have the wave equation $$u_{tt} = 4 u_{xx}$$ with the boundary conditions $$u(0,t) = u(L,t) = 0\,,\quad x \leq 0 \leq 2\pi \,,\quad t\geq 0$$ and initial conditions $$\begin{align} &u(x,0)=\...
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Appropiate Artificial Boundary Conditions for the radial part of the Klein Gordon equation?

I am trying to simulate the following equation using FDTD $ \left(- \partial^2_t + \partial^2_x + V(x) \right) \psi(x,t) =0 $ subjected to the initial conditions $\psi(x,0) = f(x),~ \partial_t \psi(x,...
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22 views

Advection diffusion equation using Crank-Nicolson with total flux and Diriclet BCs

I am trying to model the 1D advection-diffusion equation: $${\partial c \over \partial t} = D_c{\partial^2 c \over \partial x^2} -u{\partial c \over \partial x}.$$ With Robin boundary conditions that ...
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About the the stability of using an explicit scheme on the heat equation

Before I get to the heat equation I'd like to talk about the advection equation. Descritize that with FD in time and BD in space: \begin{equation} \dfrac{u^{n+1}_i - u^{n}_i}{\Delta t} + v \dfrac{u^{n}...
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1answer
101 views

How to solve heat equation in spherical coordinates with finite differences?

I have a problem dealing with heat transfer which is spherically symmetrical. I was thinking it should be possible to solve this as a 1d problem in spherical coordinates using the radius only. ...
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63 views

Solute transport around a solid obstacle

I am a newbie in CFD and single/multiphase flow and transport in general. As part of my quest to learn, I am trying to model solute transport around a solid object in the center of a 2D domain. The ...
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How to apply Neumann boundary conditions in Newton's method [closed]

Suppose that I have a very long and tedious set of differential equations. After discretization, I can get a mapping $f:\mathbb{R}^N \to \mathbb{R}^N$ such that solutions $\phi =(\phi_0,...,\phi_{N-1})...
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70 views

Instability at the boundary of a finite difference simulation of a hyperbolic PDE

I want to simulate the hyperbolic partial differential equation $$W_{tt} + V W_{tx} + k_E V W_x + k W_t = 0,$$ but I am having trouble finding a discrete analog of this equation which is numerically ...
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1answer
336 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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110 views

Computing eigenvalues of Schrodinger equation with spin

I want to solve a 2-dimensional particle in box problem with two electrons in the quantum well.I would like to take into account spin of electrons and Coulomb interactions to compute singlet and ...
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1answer
261 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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148 views

How to compute the Eigenvalue and Eigenstates of Quantum well with Effective mass using finite difference method in Python?

I want to compute the eigenvalues and eigenstates of a quantum well with different effective masses of electron in the barrier and in the quantum well. As can be seen [1]: https://github.com/mholtrop/...
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Numerical Partial Differentiation Check

In my computer vision course, we are working on extracting a 3D surface from a chain of 2D images taken under several conditions. This procedure is known as Photometric stereo. Prior to extracting the ...
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Good non oscilliatory derivatives for an exsisting grid

I'm calculating the entropy production of a shockwave by utilizing the equations: \begin{equation} \sigma = J'_q\frac{\partial}{\partial x}\left(\frac{1}{T}\right) +\frac{1}{T}\frac{4\eta}{3}\left(\...
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1answer
124 views

Implementing Dirichlet BC for the Advection-Diffusion equation using a second-order Upwind Scheme finite difference discretization

i am implementing a Matlab code to solve the following equation numerically : $$ (\frac{\partial c}{\partial t} =-D_{e} \frac{\partial^2 c}{\partial z^2} +U_{z}\frac{\partial c}{\partial z}) $$ with ...
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360 views

Why is the central difference method dispersing my solution?

I am solving numerically the ODE $\ddot x(t)=-c\dot x(t) -\sin(x(t))+F\cdot \cos(\omega t), \;\dot x(0)=x(0)=0$ for $t\in [0,20\pi]$ on an $N=2000$ dimensional grid. I am working on Python, and I ...
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218 views

(Lack of) Availability of Finite-Difference library for simple 2D PDEs

I would like to solve two types of simple 2D problems, namely the stationary heat equation on an L shaped geometry like this: And also compute the magnetostactic field in an air gap of the following ...
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1answer
119 views

Computing second derivatives with Neumann boundary condition

I am implementing a finite difference method for a PDE with a Neumann boundary condition. I will simplify my question to a single dimension. Suppose I have a PDE $$\frac{\partial u}{\partial t} = \...
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75 views

Attempt on 2d Advection with FDM - With Code

I tried to implement the 2d advection problem with a velocity field, that is not constant in space. My problem is, that the "mass" of my shifted density gets "eroded" or just ...
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27 views

shifting mass along a vector field

I have a positive matrix $\rho \in \mathbb{R}^{n,n}_+$ as a discrete probability density. Furthermore, I have a tensor $u \in \mathbb{R}^{n,n,2}$ that is constant in time and acts as a vector field. I ...
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1answer
268 views

Numerical solution of non-linear heat-diffusion PDE using the Crank-Nicolson Method

I am trying to solve numerically the following 1D EBM: $C\frac{\partial T[x,t] }{\partial t} - \frac{\partial }{\partial x}\left ( D(1-x^2)\frac{\partial T[x,t] }{\partial x} \right ) + I[T] = S[x,t](...
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1answer
90 views

Stability analysis simplification for PDE

I have the nonlinear PDE $$\frac{\partial U(z,t)}{\partial t} + A(U)\frac{\partial U(z,t)}{\partial z} + B(U)U(z,t) + C(z,t) = 0,$$ where $A(U)$ and $B(U)$ are guaranteed to be real and positive. I ...
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1answer
73 views

Derivation of a parabolic PDE using Alternating Direction Implicit method

I have a very simple question concerning Alternating Direction Implicit (ADI) Scheme. If I have an equation of the form: \begin{equation*} \frac{df(x,y,t)}{dt} = \nabla^2 f(x,y,t) + f(x,y,t) \end{...
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694 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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Recommendation for Finite Difference Method in Scientific Python

For a project I am working on (in hyperbolic PDEs) I would like to get some rough handle on the behavior by looking at some numerics. I am, however, not a very good programmer. Can you recommend ...
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101 views

CVODE Warning: Internal t = *** and h = 2.09813e-13 are such that t + h = t on the next step. The solver will continue anyway

I have a system to simulate the bubble evolution at different temperature conditions. I used CVODE_DENSE algorithm to solve the ODEs and get the bubble size and ...
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81 views

Finite Difference Approximation for the Laplacian in 2D that produces a nonsymmetric matrix

Consider the following PDE \begin{align} -\Delta u &= f \ \ \text{en} \ \ (0,1)\times (0,1) \label{P1} \\ u &= 0 \ \ \text{en} \ \partial ((0,1)\times (0,1)) \label{P2} \end{align} if we ...
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3k views

Applying Neumann boundaries to Crank-Nicolson solution in python

Consider the heat equation $$u_t = \kappa u_{xx}$$ with boundary conditions of $$u(x,0)=0\\ u(0,t)=100\\ u(l,t)=0$$ Numerical analysis by pyton can be done with ...
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1answer
68 views

Finite Difference Method on a function with multiple elements of the same array

First time posting here, so I apologize for any missing info upfront. I am working on a program in VBA that calculates a function (which itself calls another function), then calculates the derivative ...
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1answer
154 views

Why can bad jacobians sometimes works better for implicit ODE method?

I'm solving a system of stiff ODEs describing atmospheric chemistry and transport. I am using CVODE BDF from Sundials Computing. I have two ways to approximate the jacobian: Allow CVODE to ...
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TVD Lax-Wendroff with non-constant velocity

I am dealing with a linear advection equation with a non-constant velocity, where I would like to apply a TVD Lax-Wendroff scheme in 1D. The equation is the following: \begin{equation} \frac{\...
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1answer
89 views
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How to decrease error in (FTCS) forward time centered space method?

I am using the FTCS method for solving differential equations. I know that the condition for stable output is $$ \frac{\alpha \Delta t}{\Delta x ^2} < \frac{1}{2} $$ But when I use the distance ...
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1answer
159 views

Tensor product representation for the 9-point finite difference approximations for the Poisson equation

If we use 5-point finite difference approximations in a uniform rectangular grid to solve the Poisson PDE \begin{align} -\Delta u &= f \ \ \text{en} \ \ (0,1)\times (0,1) \label{P1} \\ u &= ...
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introduction to non-overlapping domain decomposition in 1D

I am new to domain decomposition and have been searching in google for an introduction in 1D which goes over the complete procedure from the continuous to the discrete problem as well as the algorithm ...
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2answers
181 views

Continuation of solution to $-\nabla\cdot (k(x,y)\nabla u)=f$

I'm trying to solve the following problem, I had previously opened another discussion for the implementation and well, it seems that it has turned out well, it can be found here. I need to calculate ...
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1answer
60 views

Compute numerical derivative for first and last points

Considering a centered finite difference approximation for the derivative, what is a reasonable approximation for the first and last points?
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2k views

Finite difference for mixed derivatives on nonuniform grid

I need to have a finite difference stencil for the mixed derivative $$f_{xy}$$ on nonuniform grids such as this one: Since I could not find a stencil in the literature, I tried to derive it by my ...
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1answer
109 views

How to find the optimum finite difference method for derivatives?

Related to: What are the negatives of using higher order finite diference schemes? Problem: I have some discrete data of a trajectory $x_t$ with errors $\delta x_t$ of a physical system sampled at ...
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356 views

Implementing routine for $-\nabla\cdot (k(x,y) \nabla u)=f$ in Matlab

I am solving the Poisson Equation for 2D given by the following expression: $$-\nabla\cdot (k(x,y) \nabla u)=f$$ in a rectangle with Dirichlet conditions on the boundary using Matlab. In principle I ...
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75 views

Solving PDEs: What is the best way to deal with non-banded/dense jacobians?

I have a system of PDEs describing atmospheric chemistry and transport. I use finite-differences to make my system of PDEs into a system of ~10,000 ODEs. I then integrate the ODEs forward in time with ...
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1answer
93 views

Calculating the Jacobian for a function containing a derivative

I have the equation $F(t) = \phi u + \frac{1}{2}\frac{d^2u}{dt^2} + u^3$ and broadly speaking, my task is to calculate the $\phi$ and $u(t)$ such that $F(t) = 0$. I am testing out a new algorithm to ...
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39 views

Expression for numerical amplification factor for Euler time integration and CD2 scheme for one degree hyperbolic function

This is the expression to be derived but I am not getting the exact expression as given in the image. Is the given expression given for the implicit Euler method? Here $N_c = c (\delta t)/h$ and $\tan(...
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2answers
100 views

Stability condition FCTS method

The FTCS method comes from the discretization of a diffusion PDE like this: $$ a^{2} \frac{u_{i+1}^{k}-2 u_{i}^{k}+u_{i-1}^{k}}{\Delta x^{2}}=\frac{u_{i}^{k+1}-u_{i}^{k}}{\Delta t} $$ If I have the ...
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2answers
179 views

why is complex step differentation better than central differences

I came across this post at Cleve's Corner, where he shows that complex step differentiation is more accurate than central differences. The error in both methods is $O(h^2)$. So why exactly does ...
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23 views

Why does this Non-Standard FDTD implementation lead to infinite increase in the magnitude of an EM pulse?

I have been working on a Particle-In-Cell Framework in Python and have noticed an issue where the magnitude of a EM pulse increasing infinitely as the simulation updates. Currently, I am using the Non-...

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