Questions tagged [finite-difference]
Referring to the discretization of derivatives by Finite differences, and its applications to numerical solutions of partial differential equations.
817
questions
3
votes
1
answer
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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 ...
- 143
2
votes
0
answers
63
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} = -\...
- 33
0
votes
1
answer
48
views
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,...
- 103
2
votes
2
answers
87
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 ...
- 23
0
votes
0
answers
38
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 ...
2
votes
0
answers
64
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
$$ \frac{\partial T}{\partial t} = \alpha \...
- 123
0
votes
2
answers
112
views
Why is this scipy.root code not converging?
I'm running a test problem to set up larger problems. Solving the simple unsteady heat equation via finite differences:
$$ \frac{\partial T}{\partial t} = \alpha \frac{\partial^2T}{\partial x^2}$$
$\...
- 123
1
vote
0
answers
44
views
How to include zero flux boundary conditions?
I am trying to solve the following differential equation in the domain of $\theta \in [0, 2 \pi]$ using finite differences scheme:
For $0< \theta \leq \pi$
\begin{align}
\rho_i^{n+1}=\rho_i^{n}+D\...
- 111
1
vote
0
answers
50
views
A staggered grid for an eigenvalue problem (linear stability analysis)
I'm interested in extending the concept of a staggered grid (commonly used to solve the incompressible Navier-Stokes equations) to a linear stability analysis context. For example, we can consider ...
- 11
2
votes
1
answer
129
views
Computing numerical derivatives
I am trying to create a sweeping surface, for which I need the frenet frame of a curve. I am trying to compute this for arbitrary curves but for testing I am just using the parametric unit half circle....
- 225
0
votes
0
answers
91
views
discrete definition of curl $ \nabla \times \vec{A}$ on a 3D grid?
I have the data for 3D vector field $\vec{A}$ (with components $\vec{A_1}$, $\vec{A_2}$ and $\vec{A_3}$) sampled on a 3D grid with integer indices i, j and k.
Assuming that only the third component $\...
- 1
1
vote
1
answer
98
views
Discretization of Poisson's equation with 2d permittivity tensor
I have to discretize a generalized Poisson equation in 2D which is
$$\nabla\cdot(\varepsilon \nabla \phi )=-\rho$$
My problem is that here $\varepsilon$ is $2\times2$ permittivity tensor
where
$$\...
- 11
0
votes
0
answers
70
views
2D wave equation is numerically unstable using Finite Difference Method
I'm working with simulating both the heat and wave equation in 2D in a Python code. When simulating the heat equation, I learned about the CFL which I used to get a numerical stable solution.
I found ...
- 101
1
vote
1
answer
110
views
Non-Uniform Grids: Approximation Quality: First Order Finite Difference vs. First Order Finite Volume
Consider the advection/transport equation in 1D with constant velocity $a(x) \equiv 1$
$$u_t(t,x) + u_x(t,x) = 0$$
on a, say, periodic domain.
On uniform grids $$ \{x_i\}_{i = 1, \dots, N}, \quad x_{i ...
- 829
4
votes
1
answer
105
views
unconditionally stable schemes better than conditionally stable ones in accuracy?
Let's consider two finite difference schemes for PDEs/ODEs. One is conditionally stable, the other is unconditionally stable. People always prefer unconditionally stable ones to conditionally stable ...
- 207
2
votes
0
answers
61
views
Compact Finite Differences for the Heat Equation with Robin Boundary Conditions
I am trying to solve the Heat equation with Robin Boundary condition:
$$ u_t(x,t) = u_{xx}(x,t), \\ u(x,0) = g(x), \\ u(0,t) + u_x(0,t) = h_0(t), \\ u(1,t) + u_x(1,t) = h_1(t)$$
for $ 0\leq x\leq1$ ...
- 21
1
vote
1
answer
76
views
Simulating First Order Hyperbolic PDE with Finite Difference Scheme
I am trying to simulate a hyperbolic PDE with some control given by the following:
$$u_t(x, t) = u_x(x, t) + \theta(x) u(0, t)$$
with boundary conditions:
$$u(1, t) = U(t) = \int_0^1 k(1-y) u(y, t)dy$$...
- 31
0
votes
1
answer
69
views
Choice of grid generation for FDM discretisation methods
I'm currently revisiting some FDM schemes for convection-diffusion equations in 1D, 2D and 3D and getting up to speed with the industry-standard methods again. The application is derivatives pricing, ...
- 31
0
votes
1
answer
127
views
Nonlinear Robin boundary condition involving square root
If you have a nonlinear second-order boundary value problem where the domain of the problem is $x \in [a,b]$, the boundary conditions imposed are the Robins condition at $x=a$ and the Dirichlet ...
- 153
1
vote
0
answers
99
views
Solving PDE on a non-uniform grid with Crank-Nicolson scheme
I am solving a 1D diffusion-type equation with the finite-difference Crank-Nicolson (CN) scheme, and I need to densify the spatial grid around the central point. One could change the spatial variable ...
- 21
1
vote
0
answers
60
views
Finite-difference produces a derivative off by one order
I have a nonlinear 2nd order boundary value differential equation where I used finite-difference method (central finite-difference) to solve it.
$$z''(x)-\frac{\frac{1}{100} z(x)^4 \left(2 z'(x)^2+12\...
- 153
0
votes
0
answers
79
views
Best approach to solve this system of equations?
I have the following 1D (in space, that is) system of equations I would like to solve:
\begin{equation}
\rho_{fs}\frac{\partial x_{fs}}{\partial t} = h_m\left(W_a - W_{fs}\right) - D_{eff}\left(\...
- 9
1
vote
2
answers
114
views
Calculate the derivative of the finite-difference method result
I have a nonlinear boundary ODE,
$$y'' + 3 y y' = 0, \qquad y(0) = 0, y(2)=1 $$
I want to solve this using the finite-difference method. I obtained the result for the data set of $y(x)$ (including the ...
- 153
0
votes
0
answers
90
views
Solving compressible euler equations in non-conservative form
I am trying to solve the following compressible 1D system of equations in two non-conservative form PDEs for $\rho, Y$ and one ODE for $v$
\begin{align}
\rho_t+(v+Q)\rho_x &= -\rho q(Y,T,\rho)-...
- 11
1
vote
1
answer
65
views
blown up solution for linear advection in upwind method with finite difference
I am going to solve this advection equation regarding the flow simulation of an energy tower
\begin{equation}
Y_t+ v(t) Y_x =0
\end{equation}
with the following boundary conditions which depend on ...
- 11
0
votes
2
answers
106
views
Practical implementation of the discrete compatibility condtion
I've recently started looking into writing a finite-differences-based solver for the Poisson equation of the form
$$\nabla\left(\varepsilon\nabla\varphi\right)=\rho$$ in 2D for arbitrary geometries (...
- 53
0
votes
0
answers
57
views
Can I use MOL to solve 2D steady state PDE in terms of r and z spatial coordinates?
recently I need to solve a 2D steady state PDE equation.
It’s not time dependent, and the only two independent variables are z and r direction.
So far for this solution, I was thinking using Method ...
- 1
2
votes
2
answers
114
views
Poisson equation with discontinuous variable coefficient
Trying to solve numerically the 2D Poisson equation with variable diffusion coefficient $K$, that in general can be discontinuous.
$$ \frac{\partial}{\partial x} ( K(x,y)\frac{\partial C}{\partial x}) ...
- 435
0
votes
1
answer
108
views
How to correctly plot Order of accuracy for different finite difference schemes
I have implemented Upwind, Lax, Lax-Wendroff, Leapfrog and macCormak method for the linear advection equation with Dirichlet boundary conditions. I am trying to create the order of accuracy plots for ...
- 1
3
votes
0
answers
113
views
Simplest way to "monotonize" MacCromack method
The MacCormack finite-difference "predictor-corrector" method is well known to generate spurious oscillations near solution discontinuities such as shock waves in gas dynamics equations. Or ...
- 337
0
votes
0
answers
42
views
Unstable FDTD for axisymmetrical wave propagation
I am writing a 2.5D axisymmetric FDTD solver for Maxwell equations. It operates on three fields on staggered grids ($E_r$, $E_z$, $B_\varphi$).
As a test problem, I am considering a cylindrical ...
3
votes
1
answer
130
views
ON the Kronecker product form of the laplacian matrix
It's well known that if we use 2nd order, centered, finite differences for the Laplace operator, we have that the matrix can be written as $$K=I \otimes A + A \otimes I $$ where $I$ is the identity ...
- 211
1
vote
1
answer
164
views
Convergence stall when solving 2D Poisson PDE with pure Neumann boundaries (finite differences)
I recently started coding a small library of 2D PDE solvers (time dependent and time independent), and my first attempt was a 2D Poisson equation of the form:
$$\nabla(\epsilon\nabla\varphi)=\nabla\...
- 53
0
votes
0
answers
69
views
Solving a system of PDEs with an ODE
I want to solve the following system of equations which consists two PDEs and one ODE:
\begin{align}
\rho_t+v\rho_x &= 0; \newline
Y_t+vY_x &= 0 ;\newline
v_t &= -\frac{1}{(\...
- 11
0
votes
1
answer
157
views
What's Kane S. Yee who invented FDTD in Chinese?
I'm not sure if the question suits this section of StackExchange, but I think the chance to get the answer is highest here (compared with other forums). So I hope more tolerance could be shown towrad ...
- 302
4
votes
1
answer
170
views
Does DCT diagonalize the FD discretisation of the Laplacian with Neumann boundary conditions?
If one has the Poisson problem (assume $\int_{\Omega} f = 0$ and $\int_{\Omega} u = 0$):
\begin{alignat}{3}
\Delta u(x) &= f(x), &\quad&x\in\Omega \\
\partial_nu(x) &= 0, &\quad&...
- 621
1
vote
1
answer
59
views
How to solve advective equation with source term depending on variable
I have the following equation
$$
\dfrac{\partial s}{\partial t} + \nabla \cdot \left( \vec{v} s\right) = f(s)
$$
Where $f(s)$ is an explicit source term that depends on $s$, e.g., ($\sin(s)\;cos(s)$).
...
- 21
4
votes
0
answers
83
views
Solving simplified 1D plasma fluid equations with finite difference
The following two equations represent a simple model of a plasma where ions are immobile.
$$n\frac{\partial u}{\partial t}+nu\frac{\partial u}{\partial x}=n\frac{d\phi}{dx}-\theta\frac{\partial n}{\...
- 141
2
votes
1
answer
680
views
How do you handle the singularity in polar or cylindrical coordinates?
Governing equations in polar or cylindrical coordinates often have terms with $\frac{1}{r}$ involved. At $r = 0$, such terms blow up to become a "singularity." The Cartesian version of such ...
- 29
0
votes
1
answer
80
views
finding discretization error in Burger equation
I was reading the paper given in the link http://www.unige.ch/~hairer/preprints/parareal.pdf and I have a problem in understanding in page 10 for the Burger equation on implementing Parareal method ...
- 41
4
votes
1
answer
163
views
How can I check mass conservation when solving the advection equation using an upwind scheme?
My question is how to keep track of the "mass" being advected out of a model domain, for the 1-D advection equation, and an upwind differencing scheme. Following is the background
Consider ...
- 297
3
votes
0
answers
58
views
Dense factorization specialized for RBF-FD method
In RBF-FD methods (see Fornberg & Flyer. A Primer on Radial Basis Functions with Application to the Geosciences. SIAM, 2015. Chapter 5.), the finite-difference stencil coefficients for a set of ...
- 342
0
votes
0
answers
51
views
How to define a stretched coordinate perfectly matched layer (PML) parameters for maximum absorbtion?
I have written a MATLAB code for solving Maxwell's equations (electromagnetic wave propagation) in 3D with perfectly matched layer (PML) boundaries. I am using a stretched coordinate PML, but I see ...
1
vote
0
answers
37
views
Calculating the species mass consumption from implicit reaction-term in diffusion-reaction equation
The 1D diffusion equation with a chemical source term has the following form:
$$\frac{\partial Y}{\partial t} = D \frac{\partial^2 Y}{\partial x^2} - k Y,$$
where $Y$ is the molar concentration of the ...
- 11
1
vote
0
answers
31
views
How does Tannehill impose boundary conditions when coding the Parabolized Navier Stokes on an Implicit Finite Differences Scheme?
I'm trying to implement the scheme he describes on his book "Computational Fluid Mechanics and Heat Transfer" on Chap.9 and I'm having trouble imposing BC.
I don’t get how he imposes them. I ...
- 11
2
votes
1
answer
117
views
"A posteriori" estimates for finite difference methods
Suppose I have a PDE in a rectangular domain that I am solving numerically via a finite difference method. How do I answer the question, "How fine do I need to make the grid to so that my ...
- 105
2
votes
1
answer
258
views
Numerical solution of 2D wave equation using Fourier transform and finite differences
This is the $2$-dimensional wave equation
$$ u_{tt} = u_{xx} + u_{yy} $$
with initial condition $u(x,y,0)=f(x,y)$ and $u_{t}(x,y,0) = 0$.
The inverse Fourier transform used is
$$ u(x,y,t) = \iint \hat{...
- 121
1
vote
1
answer
222
views
A question on the Poisson equation with Neumann and periodic boundary conditions on a rectangular region
I am trying to solve the following PDE by using finite difference
\begin{eqnarray*} \Delta u&=& f ~~on~~(0,1)\times(0,1)\\
\frac{\partial u}{\partial y}(x,0)&=&0=\frac{\partial u}{\...
- 143
1
vote
0
answers
149
views
Closed (Robin) boundaries in advection-diffusion equation with FDM
I am solving the equation
$$
\frac{\partial \phi}{\partial t} = \frac{\partial}{\partial x} \left( D \frac{\partial \phi}{\partial x} + v\phi \right)
$$
using finite differences. I want to include ...
- 11
1
vote
1
answer
113
views
constructing a symmetric matrix for finite difference
I come across the following operator in a paper
$\mathcal{I}\psi = \psi_{xxxx} + (r~\psi_x)_x$,
where $\psi=\psi(x)$ and $r=r(x)$. Periodic boundary condition is employed. It claims that the operator $...
- 217