Questions tagged [finite-difference]
Referring to the discretization of derivatives by Finite differences, and its applications to numerical solutions of partial differential equations.
700
questions
7
votes
1answer
232 views
Non-hermitian discretizations in quantum mechanics
Consider the Schroedinger equation
$$\left(-\frac12\frac{\partial^2}{\partial x^2} + V(x) \right) \psi(x) = E \psi(x)$$
The usual way to solve it is to introduce a discretization of $\psi(x)$. This ...
1
vote
0answers
123 views
Finite difference methods for coupled 2nd order nonlinear pdes
I have a system of coupled nonlinear PDEs that I cannot figure out how to solve in a smart way using FDM, so I was hoping someone here might have a clue.
The equations go as:
\begin{align*}
\frac{1}{...
0
votes
2answers
139 views
Conservative formulation for compact finite difference schemes
At the Section 4.2 of this paper (which is very well known in the computational fluid dynamic community), the author claims that it is enough, for the compact finite difference formulation in eq. 4.2....
2
votes
1answer
62 views
Discretisation of logarithmic derivative: Deriving the formula
I'm reading a paper where they use a discrete approximation of a logarithmic mass growth rate as follows:
$$ \frac{d \log M}{d \log t} \approx \frac{(t_B + t_A)(M_B - M_A)}{(t_B - t_A)(M_B + M_A)}$$
...
1
vote
0answers
41 views
Limit to volume change in a discretized mathematical model?
I have set up a mathematical model describing the diffusion of ozone out of a gas bubble. The bubble is surrounded by a thin gas film. So actually, the model describes the diffusion of ozone through ...
1
vote
1answer
1k views
second derivative with non-uniform spacing
I am trying to derive the formula for the second-order second derivative of the function $f(z)$ in the case of non-uniform spacings.
I start by considering that, around $z=\zeta_k$:
$$f(z)=f(\zeta_k)...
2
votes
1answer
125 views
normal derivatives where normal vector is ill-defined
I have to calculate the normal derivative of a function $f(i,j)$ on a domain with an irregular boundary. Let's say something like this:
x x x x
0 x 0 0
0 0 0 0
...
3
votes
0answers
73 views
Solving numerically a linearized system of elliptic (?) Navier-Stokes equation (Shallow Water Derived)
For my PhD Thesis, my advisor asked me to build a solver inspired from the article "Optimal Control Theory Applied to an Objective Analysis of a Tidal Current Mapping by HF Radar, J-L Devenon, 1989". ...
2
votes
1answer
192 views
Closed boundary conditions in finite difference method for diffusive-advective equation
I am implementing a finite difference method in solving the diffusive-advective equation:
$$
u_t + v \cdot u_x = D\cdot u_{xx}
$$
(v, D are constants). Planning to use the operator splitting method (...
1
vote
0answers
204 views
Neumann boundary condition FD implementation for instationnary diffusion equation
I am trying to solve this diffusion equation :
$\dfrac{\partial D\dfrac{\partial f}{\partial x}}{\partial x}+S = \dfrac{\partial f}{\partial t}$ ($D$ is not constant and varies according to $x$) with ...
1
vote
0answers
246 views
Global truncation error behavior at fixed time step
I am trying to solve the following diffusion equation problem:
$\frac{\partial f}{\partial t}=\frac{\partial (D\frac{\partial f}{\partial x})}{\partial x}+S$
$D=1+x^{2}+\sin(x)$
$f(x,0)=1 , f(0,t)...
3
votes
2answers
956 views
Implementing no-flux boundary condition reaction-diffusion PDE
I'm having trouble figuring out how to implement boundary conditions for this problem:
\begin{align}
\frac{\partial n}{\partial t} &= D_n\nabla^2n - \nabla\cdot\left(\frac{\chi}{1+\alpha c}n\nabla ...
1
vote
3answers
119 views
Discretization Error amplification instead of stagnation to machine precision
I wrote a code on Python 2.7.5 to solve numerically the following differential equation.
$\frac{\partial^2f}{\partial x^2}=-S$
$S=\pi^{2}\sin(\pi x)$
S is chosen that way in order to have $f= \sin(\...
2
votes
0answers
77 views
Finite differenced eigenvalue prob. of inhomogeneous boundary conditions?
I am basically asking about eigenvalue problems of differential equations using some finite difference method (FDM). Usually the system is subject to some boundary conditions (BC), e.g., Dirichlet or ...
3
votes
1answer
2k views
Applying Neumann boundaries to Crank-Nicolson solution in python
Consider the heat equation
$$u_t = \kappa u_{xx}$$
with boundary conditions of
$$u(x,0)=0\\
u(0,t)=100\\
u(l,t)=0$$
Numerical analysis by pyton can be done with
...
1
vote
1answer
98 views
finite differences on a slanted grid — advection diffusion equation
I used pretty much all my expertise with finite differences to solve an advection-diffusion equation with space-dependent coefficient with a grid in the $x$,$z$ domain with regular spacings. Something ...
3
votes
1answer
202 views
Numerical Lax-Wendroff scheme order of convergence on Burgers equation
I was suggested to move that question here.
The question to be as follows.
Statement of the problem
Is it possible to achieve the second order of convergence (OOC) of Lax-Wendroff (LxW) scheme ...
1
vote
1answer
55 views
Problems with deriving an equation for a finite-difference scheme given in the journal paper
I'm reading this paper and trying to follow everything that the author has done.
A Comparative Study of Finite Volume Method and Finite Difference Method for Convection-Diffusion Problem
But there ...
0
votes
1answer
49 views
Finite difference time domain and dynamic permittivity
Since the permittivity of any material is usually complex function of temperature, frequency, density, etc. I was wondering if it is possible to use a dynamic permittivity which changes as a function ...
0
votes
1answer
178 views
Unphysical Behaviour Characteristic-Wise WENO5-Z
I am currently working on a scheme that uses finite differences WENO5-Z with 3rd Order Runge-Kutta time integration for solving the Euler equations. The code projects the conserved variables and ...
0
votes
1answer
55 views
Trying to solve a wave-like equation
I'm trying to solve an equation whose solutions I know are plane waves but there are a few nuances.
First, the equation is of the form
$$ \partial^2_t \psi + A(r)\partial^2_r \psi +B(r) \partial_r \...
3
votes
0answers
108 views
How to account for the interface between two different phases in a discretized diffusion model?
I have tried to set up a model for the diffusion of a gas into a liquid. The two media are next to each other and the geometry is spherical because the system should simulate the diffusion out of a ...
1
vote
2answers
363 views
Mass conservation in 1d diffusion by method of lines
I am solving the 1D diffusion equation by discretization using the method of lines. My problem is that I don't manage to ensure mass conservation. I have read many similar questions about the topic ...
4
votes
0answers
197 views
How can I solve the wave equation for a circular rod in cylindrical coordinates using finite differences?
I have a problem with the stability of finite difference method for the wave equation in cylindrical coordinates.
the equation is:
$$
\frac{\partial^2 \omega_n}{\partial r^2}+\frac{1}{r}\frac{\...
0
votes
1answer
117 views
Multi-point axisymmetric boundary condition for Euler equations
I'm solving 2D axisymmetrical Euler's equations in conservative form:
$$
\frac{\partial U}{\partial t} + \frac{\partial F(U)}{\partial x} + \frac{\partial G(U)}{\partial r} = H(U)
$$
where
$$
U = \...
3
votes
1answer
747 views
Finite difference method basic implementation on Octave
Trying to study the error of FDM for a second order derivative versus step size I calculated the coefficients and validated them, but the output has errors for small step sizes.
The function in ...
2
votes
2answers
170 views
Find classical solution of transport equation with FDM
We know the classical solution of transport equation is determined by one initial (boundary?) condition, for example, the solution of
$$\frac{\partial u(t,x)}{\partial t}+\frac{\partial u(t,x)}{\...
1
vote
1answer
206 views
Finite difference method for the electric field of the electron gun
Can anybody help me to find books or MATLAB code examples for solving electric field of the electron gun(fig.1)with finite difference method? Python code examples are also perfect.
The electron gun ...
0
votes
2answers
214 views
Finite difference for computing gradients at face in finite volume code
I am working on a 3-D problem using the finite volume method (FVM). I came across a problem dealing with the computation of gradients at a face.
The geometry that I'm working with is discretized ...
0
votes
2answers
73 views
Finite difference for 2nd order ode $y'^2+y y''+\frac{2}{x} y y' -0.1 y^2=0$ with $y'(1)=0$ and $y(1)=1$
How to solve second order non-linear ODE
$$y'^2+y y''+\frac{2}{x} y y' -0.1 y^2=0$$ subject to $y'(1)=0$ and $y(1)=1$ over the interval $0 < x \le 1$.
I turned the equation to a PDE $y'^2+y y''+\...
1
vote
2answers
351 views
Conservation violation in axisymmetric Diffusion Equation
1d diffusion equation
Integrating the diffusion equation,
$$
\frac{\partial u}{\partial t} = D \frac{\partial^2 u}{\partial x^2},
$$
with a constant diffusion coefficient D using forward Euler for ...
2
votes
1answer
149 views
Combine Hydrodynamics and Electromagnetics
Is it possible, in general, to combine hydrodynamical motion and expansion of material with, say, a finite difference time domain method to simulate light-matter interaction?
If so, how is this done ...
1
vote
0answers
101 views
(Approximate) Incremental Projection Method for Navier-Stokes equations
I am trying to implement an incremental projection method for the 2D incompressible Navier-Stokes. The type of projection method I am trying is
$$
\frac{u^{*} - u^{n}}{dt} = - \nabla p^{n} - u \cdot ...
6
votes
2answers
212 views
Why naively chopped finite difference matrix works for different ODE boundary conditions
We know finite difference method (FDM) can replace $y''(x)$ as $\frac{1}{h^2}[y(x+h)+y(x-h)-2y(x)]$ or so. One naive way to write down the matrix of the differential operator is like the following, ...
3
votes
1answer
97 views
Computational Fluid Dynamics: Question on a third-order accurate finite difference approximation
According to this paper the following finite difference approximation is third-order accurate:
$$\frac{d\rho_j}{dx}\approx\frac{2-\eta}{3}\frac{\rho_{j+1/2}-\rho_{j-1/2}}{\Delta x}+\frac{1+\eta}{3}\...
1
vote
1answer
114 views
How to treat non-linear term in finite difference solution of $T''_x+T''_y+aT^2=0$?
Can we linearize $T^2$ When solving $T''_x+T''_y+aT^2=0$ by finite difference?
I solved $T''_x+T''_y=0$ in Matlab using a finite difference explicit scheme. But when there is a source term, I come ...
2
votes
1answer
89 views
Is it possible to solve multi-component Euler equations with finite-difference WENO methods?
Single-component Euler equations are solved with finite-difference WENO methods very well. Now I'm trying to apply them to gas mixtures (with aim to reacting mixtures).
While searching for extension ...
2
votes
2answers
176 views
Fixing catastrophic cancellation in velocity formula
In Writing Scientific Software: A Guide to Good Style, several disasters are mentioned including a missle defence system which led to deaths at a US base in Saudi Arabia in 1991.
The error was caused ...
2
votes
1answer
391 views
Advection equation in 2D using finite differences - the scheme works, but the pulse loses “energy”
I am trying to solve the following equation
$ \partial_t g(x,y,t) = - v\left(\partial_x + \partial_y\right) g(x,y,t)$
using finite differences (here $v>0$). The equation is also solvable ...
0
votes
1answer
312 views
Crank-Nicolson method and mixed derivatives
I am curious if anyone had literature references or knowledge on how to apply the Crank-Nicolson (with approximate factorization) to the
$$
\nabla \cdot (\nu (\nabla \mathbf{u} + \left(\nabla \mathbf{...
0
votes
1answer
56 views
Chebychev Polynomial derivatives at zero points and extreme points
I was looking for some help with derivatives of Chebychev polynomials at zero points. The recursive expression,
$$
T_{(j+1)}(x) = 2xT_j(x) - T_{(j-1)}(x)
$$
has the derivative
$$
T'_{j+1}(x) = 2T_j(...
7
votes
1answer
217 views
finite difference : why should we solve linear equation at each step
I'm trying to simulate the growth of a brain tumor using a 3d reaction-diffusion model $ \partial_{t}u = \nabla.{(D\nabla{u}) } + ku.(1-u)$ , knowing the initial distribution of tumor $u^0$, the non-...
-2
votes
1answer
302 views
Adding Non-Linear source term to 2d Implicit MATLAB code
I'm running out of time for this code so any help would be greatly appreciated. I am currently coding the 2D heat/diffusion equation in matlab but i'm having trouble adding in the source term. my ...
3
votes
2answers
398 views
Finite Difference and Finite Volume as special cases of Finite Element
I have heard this many times that "FD is a special case of FE" and separately, that "FV is a special case of FE".
However I have never come across a brief document that substantiates that claim, ...
1
vote
1answer
339 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(...
1
vote
1answer
77 views
finite difference for a second order ode
I saw in a code for discretization of something like $\frac{d^2T(x)}{d^2x}$ , ( $x = sin(\theta)$ ) tries
...
2
votes
1answer
218 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 ...
3
votes
0answers
466 views
Neumann-Neumann boundary intersection
I am trying to solve a Vertex Centered, Finite Difference, Poisson equation $-\nabla^{2}u=f$ with Dirichlet boundaries on the left and bottom and Neumann boundaries on the top and right using ...
6
votes
1answer
189 views
Fourier characteristics of repeated numerical derivative
Background
I am trying to analyse fourier characteristics of a derivative. For example if I have a first order derivative approximated as following:
$$\frac{\partial \Psi(x)}{\partial x} = \frac{\...
7
votes
4answers
569 views
Why do I still obtain a unique solution with one-sided formula when b.c. isn't enough?
Let me illustrate the issue with an simplified example. Suppose we want to solve the following problem with finite difference method (FDM):
$$\frac{\partial u(t,x)}{\partial t}=\frac{\partial^2 u(t,x)...
