Questions tagged [finite-volume-method]
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48 questions
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28 views
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 ...
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14 views
Intuition about smoothness indicator in a Flux Limiter scheme
I have a question about the choice of "smoothness indicator" when implementing a Flux Limiter scheme. I am solving a simple linear advection equation, $\left(\frac{\partial}{\partial t} + v \...
0 votes
2 answers
122 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(...
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32 views
Convergence analysis on semi discretized finite volume scheme
I am interested in studying the convergence of a semi discretized (in space) 1-D transport equation, using finite volume formalism. Let $f(t,x)$ be the solution at a time $t\in[0,T]$ and at a position ...
user1622986
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57 views
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} ...
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1 vote
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44 views
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 ...
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1 answer
105 views
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
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1 answer
35 views
Derivative on the boundary and finite volume
I am doing a solution of a PDE, and I'm solving it using the finite volume method. I have a Robin condition and I need to approximate the derivative on the boundary. Naively I write: $$\frac{df}{dx}\...
- 21
2 votes
1 answer
210 views
Spectrum of 4th order discretized Laplace operator
Consider the Laplace operator over a 1D domain, with homogeneous Neumann boundary conditions. I have discretized this operator using the cell-centered Finite Volume method on a uniform grid of size $h$...
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2 votes
0 answers
120 views
Did I get Godunov's scheme right?
I want to implement Godunov's scheme in order to simulate the nonlinear LWR-type equation $$ \partial_t u + \partial_x (u(1-u)) = 0, \quad u(0, \cdot) = u_0. $$ The update step is ($n$ denotes time ...
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1 vote
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104 views
Numerical PDEs: Is there a matrix form for solving Finite Volume methods similar to the differentiation matrix methods used for Finite Differences
I am watching some wonderful videos by Qiqi Wang on numerical methods for solving PDEs. He first introduced Finite Difference methods, where he setup the differentiation matrices for first and second ...
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2 votes
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61 views
Finite volume scheme: Is this a valid step?
Suppose we have two functions $f_1=f_1(t,x,v)$ and $f_2=f_2(t,x)$ and we do a finite volume scheme over control volumes $V_j\times X_i$ with respect to $(v,x)$. In time, one uses an implicit Euler ...
- 595
3 votes
0 answers
101 views
Branch of math similar to linear algebra but with vector exponentiation
I use Porous medium equation $$\frac{\partial u}{\partial t}=\Delta(u^m)$$ to model gas permeation through membranes. Deep down in the rabbit hole, using FVM on 1D system, I managed to derive ...
- 97
0 votes
1 answer
36 views
How was this time-average evaluated in the context of spatially second-order-accurate finite-volumes schemes?
In the context of the advection equation of the form $$\frac{\partial \phi}{\partial t}=-u\frac{\partial \phi}{\partial x},$$ the author in [1] discussed how many schemes are based on second-order ...
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2 votes
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26 views
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} \\ \...
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