Questions tagged [finite-volume-method]
The finite-volume-method tag has no summary.
48 questions
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How to implement a source term on the boundary of a finite volume (or discontinuous galerkin) method?
I've written a 3D linear acoustic discontinuous galerkin method that works on an unstructured tetrahedral mesh. This models pressure, $p$ and velocity, $\mathbf{u}=(u,v,w)$ and has fully reflective ...
0
votes
2
answers
119
views
Does (homogenous) Dirichlet boundary condition kills consistency in finite volume method?
I am currently working on the following transport equation :
\begin{align}
\frac{\partial}{\partial t}f(x,t) + \frac{\partial}{\partial x}\left[v(x)f(x,t)\right] &= 0, \\
v(0)f(0,t) &= 0, \\
f(...
- 11
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55
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What does Godunov's scheme actually converge to?
So assume we have a scalar conservation law
$$
\partial_t u(t, x) + \partial_x \big(f(x, u(t, x))\big) = 0 \text{ in }(0, T) \times \mathbb{R} \quad \quad u(0, \cdot) = u_0
$$
for some $u_0:\mathbb{R} ...
- 6,939
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1
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99
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How do we know when total variation diminishing (TVD) schemes are appropriate to apply to a given PDE?
I am working on the research of hydrodynamics and see some researchers applying the TVD numerical scheme to flow equations such as shallow water equations or kinematic wave equations. But in their ...
- 33
2
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1
answer
205
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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 $h$...
- 91
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0
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103
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Numerical PDEs: Is there a matrix form for solving Finite Volume methods similar to the differentiation matrix methods used for Finite Differences
I am watching some wonderful videos by Qiqi Wang on numerical methods for solving PDEs. He first introduced Finite Difference methods, where he setup the differentiation matrices for first and second ...
- 2,705
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100
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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 ...
- 97
2
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0
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25
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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} \\
\...
- 2,408
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649
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confusion between finite difference methods and finite volume methods for PDEs
I am new to numerical methods for PDEs, but I am seeing some confusing perspectives in two different common textbooks: Langtangen's book on Finite Differences and Leveque's book on Finite Volume ...
- 2,705
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104
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Why the inlet and outlet fluxes are different when I solve the Poisson equation by FEM over a trapezoidal domain?
I solved the Poission equation which is given by
\begin{equation}
\Delta h = 0,
\end{equation}
where $h$, in my case, is the hydraulic pressure. Because I want to solve steady-state flow, the $\nabla ...
- 161
1
vote
1
answer
52
views
What cone is the one I should use?
I have this region:
$S={(x,y,z) \in \mathbb{R}^3: x^2+y^2+z^2\leq 1 \wedge z^2\geq3(x^2+y^2) \wedge z \geq 0} \nonumber$
I need to determine the volume but I don't know which cone should I choose.
...
- 95
5
votes
1
answer
381
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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,...
- 1,097
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47
views
Change of variable of surfaces integrals
Let $\mathbb{S}^{2}$ be the unit sphere, $\Delta_{s}$ the Laplace-Beltrami operator, $U_h$ the finite element space in $\mathbb{S}^{2}$ and $V_{h}$ the space of constant function associated to $U_{h}$....
- 11
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1
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307
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Using Runge Kutta in local lax friedrichs fvm for shallow water problem
This question is working on the same problem but using a new concept as mentioned here:.
Now I am trying to expand this concept using the Runge kutta (RK4):. Now the original PDE is as follows:
$$\...
- 81
1
vote
1
answer
165
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Simplification of a volume integral
While trying to simplify an equation in the integral form (in the context of finite volume method with polyhedral meshes), I came across this term:
$$
\iiint_V{\vec v\nabla\cdot\vec c\,\mathrm dV} \...
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0
answers
181
views
find centroid of hyperpyramid
I'm having trouble computing centroid of hyperpyramid (assume we have n points in n dimension). I already search a lot and I find how can calculate triangle and pyramid centroid, but I don't know how ...
- 113
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votes
1
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226
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Implement boundary conditions in finite-volume code for conservation laws
For the numerical solution of scalar hyperbolic conservation laws using finite volume schemes.
In order to implement the boundary conditions and the numerical
fluxes, make use of Ghost cells.
...
3
votes
1
answer
575
views
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:
$$u=\text{function of x,t}\\\hat{u}=\frac{1}{\Delta x}\int_{x_{i-1/...
- 910
2
votes
0
answers
70
views
non-consistent initial conditions in finite volume method
Assume the wave equation in two dimensions:
$$
\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}
$$
where $x$ and $y$ represent spatial variables (...
- 457
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108
views
Finite-volume method applied to a particular advection equation
I'm trying to apply the finite-volume method (FVM), with which I'm not so familiar, so a simple 1D PDE equation.
The equation I want to solve is, to simplify,
$$\frac{\partial U}{\partial t} + A\...
- 41
2
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0
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170
views
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 ...
- 827
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answers
4k
views
energy equation in cylindrical coordinates and conservative form of fluid flow equations
We can easily find the energy equation for incompressible fluid as a temperature equation:
$$
\rho c \frac{DT}{Dt}=\nabla\cdot(k\ \nabla T) + \tau_{xx}\frac{\partial u}{\partial x}+\tau_{yx}\frac{\...
- 663
1
vote
2
answers
2k
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Finding the volume-weighted average centroid of a polyhedron
In order to find the centroid of a polyhedral element (for finite volume method) bounded by a set of vertices, we follow the following procedure (from my textbook):
1 - Calculate the geometric centre $...
- 264
2
votes
1
answer
354
views
finite volume methods: what do I have to do with the cell averages after each step?
I'm having a hard time understanding finite volume methods.
If I take for example the scalar advection equation
$$\partial{u}_{t}+a\partial{u}_{x}=0, a>0$$
with suitable initial and bondary ...
- 153