Questions tagged [finite-volume-method]

34 questions
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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?
4
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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…
4
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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…
3
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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…
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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
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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…
2
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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.…
2
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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…
2
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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…
2
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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{\partial u}{\partial y} +\tau_{zx}\frac{\partial…
2
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1 answer

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 conditions, and use cell averages with $x_{i}=i\triangle x$…
1
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1 answer

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 h =[\frac{\partial h}{\partial x}, \frac{\partial…
1
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1 answer

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. Half of full cone?
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0 answers

Approximating a function by integrating it

I stumbled over this approximization for a flow-map $m:\Omega\rightarrow \mathbb{R}^2, \Omega \subset \mathbb{R}^2$ in this paper, pages 4-5. The authors approximate the map by integrating it over subsets of its domain. I dont understand, why we can…
dba
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Find the volume of the solid generated by revolving the triangular region enclosed by $y = |x|$ and $y = 1$ about the line x = −2.

Question: Find the volume of the solid generated by revolving the triangular region enclosed by $y = |x|$ and $y = 1$ about the line $x = −2$. My solution: $$\pi*9*1 - \pi*1*1 - 1/3 *\pi* 1 - 1/3 *\pi* 1 = 22/3 \pi$$ Use the large cylinder minus the…
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