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Questions tagged [finite-volume-method]

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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. ...
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Find the volume of the cylinder $x^2+y^2\leq 2$ bounded by the xy plane and $z=x^2+y^2$

Find the volume of the cylinder $x^2+y^2\leq 2$ bounded by the xy plane and $z=x^2+y^2$ Not sure how to proceed
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Spline interpolation base on average of function

Let's choose the domain as the unit square and divide it into N^2 subdomain. Now I only know the average of function on each subdomain. I know i can regard the average as the value of the function at ...
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Characteristic lengths of an hexahedron

I am working on an error estimation strategy for a finite element code working on unstructured anisotropic meshes. In order to compute this error estimator I need the characteristic lengths of any ...
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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/...
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information regarding numerics of pde

I am learning conservation laws at the moment and now I have to start numerical solutions of the pde. I have to do finite volume schemes, it is advisable to go through finite difference first and then ...
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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 (...
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How is divergence theorem used in this proof

I am studying conservation laws, i was doing the proof for the rankine hugoniot condition, in the middle of the proof they have used divergence theorem , but I do not understand how they have used ...
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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\...
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1answer
584 views

Solving transport equation with pseudo-spectral and finite volume methods in MATLAB

I have a linear transport equation $$ \dot{c}(x; t) + vc_x(x; t) = 0 \tag{1}$$ on an interval $[0; 2π]$ with $v = 1$, periodic boundary conditions and two different initial values $$c(x; 0) = \sin(x)$$...
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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 ...
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advective tem in conservative equation in cylindrical coordinates

A conservative equation for the general property $\phi$ can be written as: $$ \frac{D \rho \phi}{Dt}=\nabla\cdot(\Gamma_\phi \nabla\phi)+S_\phi $$ where $\frac{D}{Dt}$ is the material derivative, $\...
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807 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{\...
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349 views

Implementing Neumann boundary conditions for finite volume PDEs

I'm trying to better understand finite volume methods and have started coding up a basic script to solve the diffusion equation $$u_t = u_{xx}$$ which has the finite volume form: $$\frac{\bar{u}^{n+1}...
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785 views

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 ...
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1answer
778 views

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 ...
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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 ...
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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?