Questions tagged [discretization]
The process of representing a continuum space with a finite set of points/elements
0
votes
1answer
34 views
How to do Weierstrass-transform in MATLAB?
I have a diagonalization problem. I have the eigenstates correctly, and I want to do a Gaussian-smearing (Weierstrass-transform) on them. So I have the wave functions ($\Psi$), and the continuous ...
0
votes
1answer
59 views
Is “Gradient Computation” in Finite Volume Discretization Really 2nd order accurate?
Based on this, pp 245, we go through these steps to discretize a gradient statement, namely $\nabla\phi$:
1- Gauss theorem reads,
$$
\int_V\nabla \phi dV = \oint_{\partial V}\phi dS
$$
2- Integral ...
2
votes
1answer
35 views
Discretization with non-constant matrix containg entries form unknown vector
Consider a system of PDEs
$$
\begin{cases}
u_t = \nabla \cdot (D(u)\nabla u) + \frac{c}{K_U+c}u-ku\\
c_t = d_c\Delta c -\frac{\nu_U c}{K_U + c}u
\end{cases}
$$
with some boundary conditions. Here, $D(...
1
vote
1answer
66 views
Approximation Error in a Finite Difference Approximation of the Square of Derivative
First Part: (First-order derivative)
Assuming $f$ is an infinitely differential function everywhere, the Taylor series of $f(x + h)$ at $x$ is
\begin{align}\tag{1}
f(x + h) = f(x) + hf'(x) + \frac{1}...
0
votes
0answers
69 views
Analytic vs discrete understanding of PDE
The PDE I am working with:
$$\partial_tu = \nabla \cdot (a(x)\nabla u)-\beta(x)u\\
\partial_nu=0, x \in \Omega \subset \mathbb{R}^2\\
\beta(x)>0$$
Integrate the PDE:
$$\int_\Omega \partial_t u=\...
0
votes
0answers
32 views
What will PDE discretization matrix look like for time and space? [duplicate]
Please note: this question is not a duplicate of this question since, while the PDE is the same, the nature of this question is different, i.e. the other question treats a different aspect of this PDE....
1
vote
0answers
51 views
Discretizing a parabolic PDE with finite volume method
I want to discretize the following parabolic PDE:
$$u_t = \nabla\cdot(\alpha(x)\nabla u)- \beta u\\
x\in\Omega \subset \mathbb{R}^2\\
\partial_n u = 0\\
u(t,0) = u_0(x)\ge 0, \alpha(x)>0$$
Given ...
1
vote
0answers
59 views
Determine truncation error of PDE discretization
The equation is $$\frac{\partial}{\partial x}\left(u\frac{\partial u}{\partial x}\right)=f(x)\\ 0<x<1, u(0)=u(1)=0$$
I'm discretizing this PDE using FVM as follows:
$0=x_0=x_{1/2}<x_1<x_{...
0
votes
0answers
46 views
Existence and uniquness of solution of FVM for Poisson equation
I'm discretizing the following Poisson equation using FVM where the domain $\Omega$ of the solution is a regular hexagon of side $1$ centered about the origin.
$$\Delta u =k,\text{ $k$ constant}\\
\...
0
votes
1answer
74 views
Elliptic PDE finite volume method with Dirichlet boundary condition
I want to discretize the following equation using a Finite Volume Method
$$\nabla \cdot (a(x)\nabla u)=f(x)\\x\in \Omega \subset \mathbb{R}^2
\\u_{|\partial\Omega}=g$$
I'm using Voronoi cells here: ...
3
votes
2answers
110 views
Finite volume discretization of non-conservative linear hyperbolic equation
Problem. Consider the one-dimensional adjoint Euler equations for $(x,t) \in \Omega \times [0,T]$ with $\Omega \subset \mathbb{R}$ and $T > 0$
$$ \varphi_t + \Big(\frac{\mathrm{d}F}{\mathrm{d} U}(x)...
0
votes
0answers
38 views
Discrete maximum principle for discretized ODE
I discretized the following ODE using central finite differences for 1st and 2nd derivatives:
$$u''-bu'=f(u), x\in (0,1)\\u(0)=1, u'(1)=0\\ b>0, f:\mathbb{R_{\ge 0}}\to \mathbb{R}_{\ge 0}$$
The ...
1
vote
0answers
75 views
PDE discretization on triangular domain
Given the 2D Poisson equation
$$\Delta u = f\\ u(x,0) = g_1(x), 0<x<1\\u(0,y) = g_2(y), 0<y<1\\
\partial_n u (x, 1-x) =0, 0<x<1$$
defined on the domain $\Omega := \{(x,y) \in \...
0
votes
1answer
103 views
Discretization Neumann boundary condition
I'm currently working with the following Poisson equation with mixed boundary conditions, including a Neumann boundary condition.
$$\Delta u = f\\ u(x,0) = g_1(x), 0<x<1\\u(0,y) = g_2(y), 0<...
0
votes
1answer
48 views
Changing the domain of a 3D Finite Difference code from cube to sphere
I have an explicit FD (Finite Difference) code for diffusion/heat on a PDE in a cuboid domain, and it works fine. I would like to update the discretized equations and change the code so as to solve ...
4
votes
1answer
104 views
Matrix Representation of a Discretization for a Partial Differential Equation
I want to discretize the following problem
\begin{cases}
\mu \nabla^2u+(\lambda+\mu)\nabla \nabla\cdot u = \rho \frac{\partial^2u }{\partial t^2 } + \beta \frac{\partial u}{\partial t}\\
u(...
6
votes
1answer
110 views
How do I integrate a function defined over an arbitrary area?
Let's say, I have a compact area $S$ (for example a circle, a square or some arbitrary polygon) and a function $f: S \rightarrow \mathbb{R}$. I want to numerically calculate the Integral
$$ \int_S f(\...
2
votes
1answer
84 views
Stability Analysis
The partial differential equation,
\begin{align}
\dfrac{\partial f}{\partial x} + a(x)\dfrac{\partial f}{\partial y} = 0
\qquad & f(0,y) = f(L_1,y) = c_0e^{-y} \\
& f(x,0) = c_0 \;,\; f(x,L_2) ...
5
votes
1answer
175 views
Finite Differencing schemes for Convection-Diffusion equation
I'm using the Convection(/advection)-Diffusion(-Reaction) equation to calculate the temperature over time in different hydraulic parts like a pipe or a heat exchanger.
The flow/convection is always 1D,...
2
votes
1answer
72 views
Is there Von Neumann stability analysis for 9-point laplacian like we have for the 5-point Laplacian?
For spatial accuracy in 2-D Laplace equation, a 9-point stencil is better than a 5-point one.
$$\partial_tq= r\left(\partial^2_x q + \partial^2_y q\right)$$
for FTCS (forward-time, central-space) ...
1
vote
1answer
50 views
How to isolate and test time discretization order of accuracy
I have a code that uses both spatial and time discretization/integration. For convergence analysis, I am wondering how one would test the order of accuracy of their ${time}$ integration scheme? I ...
0
votes
1answer
23 views
Calculating volume of a discretised diffuse interface object
Suppose I have a spherical object projected onto a discrete square mesh. The dicretised circle can be represented by filling a logical matrix such that voxels in the interior of the sphere are filled ...
2
votes
1answer
58 views
Discretisation of logarithmic derivative: Deriving the formula
I'm reading a paper where they use a discrete approximation of a logarithmic mass growth rate as follows:
$$ \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)}$$
...
1
vote
0answers
51 views
Decrease in slope during convergence analysis
I am using the method of manufactured solutions to perform the order of accuracy testing. I am using a cube for the testing. The cube is size 1m on all sides.
I used 5 refinements:
$dx = dy = dz = ...
1
vote
0answers
38 views
Limit to volume change in a discretized mathematical model?
I have set up a mathematical model describing the diffusion of ozone out of a gas bubble. The bubble is surrounded by a thin gas film. So actually, the model describes the diffusion of ozone through ...
2
votes
1answer
88 views
FEM-Laplace with Dirichlet in only a few points: Nonsingular operator?
Let's consider the FEM discretization of the Laplace operator without boundary conditions, i.e.,
$$
a(u,v) = \int_\Omega \nabla u \cdot \nabla v - \int_\Gamma (n\cdot \nabla u) v.
$$
For one-...
5
votes
2answers
146 views
Residual norm of PDE discretization: correspondence in the continuous problem?
Solving a linear PDE like
$$
\Delta u = f \quad\text{on } \Omega,\\
n\cdot \nabla u = 0 \quad\text{on } \Gamma,
$$
with Finite Elements usually goes like this:
Create the discretization $Au=b$ via
$$
...
1
vote
0answers
82 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)...
1
vote
3answers
85 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(\...
1
vote
1answer
50 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 ...
2
votes
0answers
88 views
Monotone, monotonicity preserving, LED, TVD, NVD, bounded, stable and stability preserving discretization schemes [closed]
When it comes to discretization schemes for finite volume method, the following terms can be found in literature:
monotone schemes
monotonicity preserving schemes
local extremum diminishing schemes
...
1
vote
5answers
345 views
Don't we care about the numerical diffusion in the diffusion term?
In the context of the solution of advection-diffusion equations by finite volume method, many numerical schemes, papers and book chapters are dedicated to address the numerical diffusion and/or ...
3
votes
0answers
98 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 ...
3
votes
1answer
112 views
Discontinuous Galerkin FEM : Control points are mid-points of edges instead of nodes
I am thinking to use discontinuous galerkin FEM (DGFEM) method to estimate discontinuous displacement field $u: \Omega \rightarrow \mathbb{R}^2$ at the crack surface of a material.
The domain is ...
0
votes
1answer
40 views
Implementation of stochastic cellular automata
In my problem, I have a lattice with a stochastic cellular automaton. In order to simplify a bit, let's say it is 1D. In my system, each node can be type A, B or C. A way to represent the system and ...
1
vote
1answer
131 views
Can this nonlinear advection-diffusion equation be discretized as to only have to solve SPD systems?
Consider a nonlinear advection-diffusion equation of the form
$$
\frac{\partial u}{\partial t} = \nabla \cdot (a(u) \nabla b(u) - \vec{c}(u)u) \tag{1}
$$
on a rectangular domain with Dirichlet ...
1
vote
0answers
232 views
Implement Robin boundary condition (finite volume)
I have a PDE equation with Robin Boundary condition in an annulus system and I should solve it by finite volume method:
\begin{align}
\frac{\partial T_f}{\partial t} - k \left(\frac{\...
1
vote
0answers
80 views
Discrete operator textbooks
This will be a vague question.
When I was writing a finite element matrix assembly routine, a colleague noticed that I had a bug in my code because the sparsity pattern of the one of the blocks didn'...
7
votes
1answer
394 views
Discrete wave simulation - absorbing boundaries?
I wrote a simple 2D wave simulation using the following equations:
$$\frac{\partial^2 u}{\partial t^2}=c^2\nabla^2u$$
Where $\nabla^2$ is the discrete laplace operator using a Von Neumann neighborhood ...
1
vote
1answer
257 views
Finite Volume Polar Discretization: Lengths
Given a uniform polar grid, as in the figure below:
and a FV discretization of a gradient for example:
$\frac{\partial p}{\partial \varphi} = 0$
$\Delta r \frac{p_e - p_w}{\Delta \varphi} = 0$
My ...
2
votes
0answers
78 views
Literature on numerical solving based on multiple meshes?
Consider solving a differential equation system for 1D, 2D, or 3D.
It involves various input and output "field" variables, which, more often than not, correspond to various physical quantities ...
4
votes
0answers
301 views
Order of accuracy of FVM discretization
I've recently got interested in CFD and started a small project by solving the radial Reynoldsequation. Why the Reynoldsequation? I recently encountered it through my studies and somehow got stuck :)
...
1
vote
0answers
305 views
Rhie and Chow Pressure Velocity Coupling
In a collocated grid, which velocity is used in the convective term in the momentum equation? Is the Rhie and Chow constructed face velocity or an average of the adjacent cell center values(in a ...
1
vote
0answers
74 views
Three steps of pde numerical solution and nonlinear equation
I'm very new here.
I'm trying to solve nonlinear elliptic equation
$$
(n(u)u')' = f(u)
$$
and face with crucial misunderstanding.
As I suppose, the procedure of solving some nonlinear equation ...
-3
votes
1answer
158 views
Use Finite Difference Discretization to find approximate solution to the Poisson's equation
I've just been introduced to the Poisson's equation. I've never had the need to dealt with PDE, so I'm a bit lost.
Apparently we can compute an approximate solution of the Poisson's equation
$$\frac{...
2
votes
2answers
130 views
Combining trapezoidal rule with upwind scheme
I want to discretize and numerically solve the equation:
\begin{equation}
v(k)\dfrac{\partial f}{\partial z} + F(z)\dfrac{\partial f}{\partial k} = \alpha f(z,k) \,\,.
\end{equation}
Discretization ...
1
vote
1answer
280 views
Mass Lumping in case of Dirichlet boundary conditions
I'm currently trying to implement a FEM to solve a type of wave equation with homogeneous Dirichlet boundary conditions by using standard $\mathcal{P}_1$ triangle elements and an explicit scheme for ...
1
vote
0answers
36 views
Inner Products vs Discretizations of Functions
Let $f:(0,1)\rightarrow [0,1]$ be a $\mathcal{C}^{\infty}$ function.
We say that a function $D_r^f:\{1,...,r\}\rightarrow [0,1]$ is an $r$-discretization of $f$ if $$D_r^f(j) = \frac{1}{|a(j)-b(j)|}\...
3
votes
1answer
126 views
Space-time Galerkin of Burgers changes the convection speed
tldr: Can space-time Galerkin schemes applied to convection-diffusion problems lead to effects on the convection velocity?
For time $t\in (0,1)$ and the spatial variable $\xi \in (0,1)$, I am ...
3
votes
1answer
163 views
Motivation behind Collocation Method
In previous question "Motivation behind Galerkin method", Paul gives a good and easy-understanding explanation indicating that the Galerkin method is a kind of projection method. Can anyone explain ...
