Questions tagged [finite-volume-method]
49 questions
30
votes
7 answers
What is the difference between Finite Difference Methods, Finite Element Methods and Finite Volume Methods for solving PDEs?
Can you help me explain the basic difference between FDM, FEM and FVM?
What is the best method and why?
Advantage and disadvantage of them?
Anh-Thi DINH
- 573
5
votes
1 answer
Discrete entropy inequality for scalar conservation laws
Consider a scalar conservation law $u_t+f(u)_x=0.$
A three point monotone scheme given by,
\begin{eqnarray}
u_i^{n+1}=u_i^{n}-\lambda (F(u_i^n,u_{i+1}^n)-F(u_{i-1}^n,u_i^n))
\end{eqnarray}
where $F(u,u)=f(u).$
For a general entropy flux pair…
Rosy
- 1,087
4
votes
1 answer
Factor quadratic functions into weighted sum of squares
I am working to reproduce results from a paper [links directly to page 17, equations 3.4 on some browsers] on a finite-volume method reconstruction method called WENO which I'm using in one dimension. Part of the method involves calculating a…
Rob Smallshire
- 93
- 7
3
votes
0 answers
Branch of math similar to linear algebra but with vector exponentiation
I use Porous medium equation
$$\frac{\partial u}{\partial t}=\Delta(u^m)$$
to model gas permeation through membranes. Deep down in the rabbit hole, using FVM on 1D system, I managed to derive something similar to a state representation of the…
Lukáš Mrazík
- 97
3
votes
1 answer
deriving the differential form of a PDE using finite volume method
I am looking at Leveque's book on finite volume methods for hyperbolic problems. I understand the method, but for some reason I am having a little trouble understanding this particular algebraic manipulation below from page 17 of the book. Let's…
krishnab
- 2,653
3
votes
1 answer
Lax-Wendroff finite volume scheme derivation
I'm trying to figure out how the finite volume version of Lax-Wendroff scheme is derived.
Here is the PDE and Lax-Wendfroff scheme:
$$u=\text{function of x,t}\\\hat{u}=\frac{1}{\Delta x}\int_{x_{i-1/2}}^{x_{i+1/2}}u\thinspace dx \text{ (the average…
Frank
- 910
3
votes
1 answer
Upwind differencing scheme in Finite Volume Method (FVM)
I have some troble in understanding how I can assess the direction of the flow for the upwind differencing scheme.
Lets say we have the following ODE:
$$a(x)\phi '(x)+b(x)\phi ''(x)=f(x)$$
Now how do I asses the direction of the flow for this…
MrYouMath
- 16,174
2
votes
0 answers
Why does integrating $u\frac{\partial \phi}{\partial x}$ with respect to $x$ yield a net flux in the $1-$D advection equation for finite volumes?
When developing a finite-volume discretization of the 1-D advection equation: $$ \frac{\partial \phi }{\partial t} + u\frac{\partial \phi}{\partial x} = 0$$
Where $u$ is the advection velocity in m/s and $\phi$ is a scalar.
It is required to write…
2
votes
1 answer
Spectrum of 4th order discretized Laplace operator
Consider the Laplace operator over a 1D domain, with homogeneous Neumann boundary conditions. I have discretized this operator using the cell-centered Finite Volume method on a uniform grid of size $h$, employing a four-point central difference…
Nicola
- 91
2
votes
0 answers
Did I get Godunov's scheme right?
I want to implement Godunov's scheme in order to simulate the nonlinear LWR-type equation
$$
\partial_t u + \partial_x (u(1-u)) = 0, \quad u(0, \cdot) = u_0.
$$
The update step is ($n$ denotes time step, $j$ spatial step)
$$
u^{n+1}_j = u^n_j -…
Hyperbolic PDE friend
- 6,919
2
votes
0 answers
Finite volume scheme: Is this a valid step?
Suppose we have two functions $f_1=f_1(t,x,v)$ and $f_2=f_2(t,x)$ and we do a finite volume scheme over control volumes $V_j\times X_i$ with respect to $(v,x)$. In time, one uses an implicit Euler scheme.
Suppose, we have already done the implicit…
selector
- 585
2
votes
0 answers
Finite Volume Method Dirichlet conditions on sets of measure zero
I have a Poisson equation with mixed boundary conditions:
\begin{equation}
\begin{alignedat}{3}
-\Delta u(\vec{x}) &= f(\vec{x}), &&\quad \vec{x}\in\Omega \setminus \mathcal{K} \\
\vec{n} \cdot \nabla u(\vec{x}) &= 0, &&\quad \vec{x} \in…
lightxbulb
- 2,378
2
votes
0 answers
non-consistent initial conditions in finite volume method
Assume the wave equation in two dimensions:
$$
\begin{cases}
u_{xx}+u_{yy} = u_{tt}\\
u(x,y,t=0) = f(x,y) \\
u_t(x,y,t=0) = g(x,y)
\end{cases}
$$
where $x$ and $y$ represent spatial variables (Cartesian Coordinates) and $t$ represents time.…
Snifkes
- 457
2
votes
0 answers
Finite-volume method applied to a particular advection equation
I'm trying to apply the finite-volume method (FVM), with which I'm not so familiar, so a simple 1D PDE equation.
The equation I want to solve is, to simplify,
$$\frac{\partial U}{\partial t} + A\left(x\right) \frac{\partial}{\partial…
merrihurruz
- 41
2
votes
0 answers
What is the advantage to have a locally conservative numerical scheme?
Numerous papers tackle the issue to formulate conservative numerical schemes to solve PDEs. For example Liu, Wang, Zou claim "local mass conservation [...] is a highly preferred property of the algorithm in practical computing."
While I do…
don-joe
- 827