Questions tagged [finite-volume]
Referring to the discretization of partial differential equations using Finite Volume Method.
265
questions
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FVM for non-regular domain with triangular mesh
Setup
The 1D convection-diffusion equation is given by:
\begin{equation}\tag{1}
\frac{\partial u}{\partial t} + v \frac{\partial u}{\partial x} - \mu \frac{\partial^2 u}{\partial x^2} = 0,
\end{...
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1
answer
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Adding a diffusion term to the MUSCL - Kurganov and Tadmor central scheme
Im currently using a MUSCL scheme with a rusanov flux and Van Leer limiter to simulate the 2d euler equations:
$$ \frac{\partial \rho}{\partial t} + \frac{\partial \rho v_x}{\partial x} + \frac{\...
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3
answers
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Approximation of derivatives in the finite volume method
I am looking into the finite volume method and I have come to a problem with discritisation. Suppose I am looking at a particular cell(I'm dealing) with cartesian grid. at the points $(X_{i},Y_{j}),(...
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How is entropy taken into account in a FV solver?
When solving for a given PDE, how is the entropy taken into account? Does the fluxes has another form? Or is the entropy PDE for a given entropy-flux included in the solver?
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Symmetrization of Laplacian Matrix Operator (finite volumes)
The aim is to construct symmetric Laplacian matrix $L$ for 2D square domain.
I have the following discretization of second derivatives on non-uniform grid (will skip the steps of derivation, any ...
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0
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28
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Parallel Block-Structured class abstraction for FDM
I’m currently developing a FDM/FVM (using contravariant coordinates) code using Fortran and Co-Arrays (SIMD, in general), and so far I have all sparse matrix (BiCGStab, working on AMG) solvers and ...
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Refluxing step on Finite difference AMR
Hi I am a computer scientist working on MHD code for astrophysics simulation. We use a finite difference scheme where we first solve the spatial derivatives and with them solve the right hand side and ...
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How to find volume of a depression in a bone
This is known as an articular pillar. It lies on the outer surface of the bone. We have 3D Slicer openCV, but we are unable to find it as its a little irregular in shape and also its 3D in nature. ...
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Finite volume method using Chebyshev polynomials
I want to solve the following set of coupled advection-diffusion equations:
$$
\frac{\partial f}{\partial t}=\nabla\cdot(\kappa\nabla f)+\nabla\cdot(\boldsymbol{u}f)+s_f(g),
$$
$$
\frac{\partial g}{\...
3
votes
2
answers
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views
Non-conservative advective term in a finite volume scheme
I am interested in solving this set of nonlinear couples advection-diffusion equations using a finite volume scheme:
$$
\frac{\partial f(x,y)}{\partial t}=-(\boldsymbol{u}+\nabla\eta)\cdot\nabla f +\...
1
vote
1
answer
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Non-Uniform Grids: Approximation Quality: First Order Finite Difference vs. First Order Finite Volume
Consider the advection/transport equation in 1D with constant velocity $a(x) \equiv 1$
$$u_t(t,x) + u_x(t,x) = 0$$
on a, say, periodic domain.
On uniform grids $$ \{x_i\}_{i = 1, \dots, N}, \quad x_{i ...
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1
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0
answers
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Fortran - Lid-Driven Cavity Boundary Conditions Error when using SIMPLE method
I am studying Numerical Methods for incompressible flows. part of the tasks is to model the lid driven cavity problem in 2D using the SIMPLE method.
I have been provided with Fortran code that is ...
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0
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Exponential Integrator to solve PDE with Stiff term
I wish to solve an equation like the following,
$$\frac{\partial f}{\partial t}+\frac{\partial}{\partial x}\left(A(x)f\right)=S(x,t)f$$
where $A(x,f)f$ and $S(x,t)f$ are the advection and the source ...
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votes
1
answer
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Numerical solution of an advection equation, $\frac{\partial P}{\partial t}+\frac{\partial}{\partial x}\left(P^{5/3}\right)=0$, with finite volume
I was trying to solve the following equation numerically,
$$\frac{\partial P}{\partial t}+\frac{\partial}{\partial x}\left(P^{5/3}\right)=0$$
I adopted the Godunov approach for discretising the ...
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votes
1
answer
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Gmsh Python: Specify mesh regularity conditons
I am using python API of Gmsh to generate a mesh for a rectangular domain. I am really new at this. My code looks like this,
...
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0
answers
82
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(Algorithmic) Differentiation capable Finite Volume Software: Generation Jacobian
I am looking for Finite Volume Software that employs a method-of-lines like approach by constructing from the hyperbolic PDE of form
$$\partial_t \boldsymbol u(t,\boldsymbol x) + \nabla \cdot \...
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0
answers
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Navier Stokes Equation Discretization by upwind scheme
I've been reading this paper regarding the SIMPLER algorithm in CFD and I don't understand exactly how the discretized equation mentioned by the author is arrived at. It says that the convective terms ...
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0
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Parallel Plate Flow simulation using SIMPLER Alogortithm
I am new to the community and CFD itself and am working on my first CFD program in Scilab based on the SIMPLER algorithm in Scilab i.e. modeling a flow between parallel plates with entrance region. I ...
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Can I use MOL to solve 2D steady state PDE in terms of r and z spatial coordinates?
recently I need to solve a 2D steady state PDE equation.
It’s not time dependent, and the only two independent variables are z and r direction.
So far for this solution, I was thinking using Method ...
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1
answer
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How to solve advective equation with source term depending on variable
I have the following equation
$$
\dfrac{\partial s}{\partial t} + \nabla \cdot \left( \vec{v} s\right) = f(s)
$$
Where $f(s)$ is an explicit source term that depends on $s$, e.g., ($\sin(s)\;cos(s)$).
...
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answers
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Centered finite volume scheme for an advective term on an unstructured/irregular/non-uniform grid
Consider the continuity equation
$$\frac{\partial u}{\partial t} + \frac{\partial \Phi}{\partial x} = 0$$
$$\Phi = au + b\frac{\partial u}{\partial x}$$
Suppose I want to solve the above using ...
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1
answer
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Computing material derivated of tensor quantity
I would like to compute the material derivated of a tensor quantity, in the context of the finite volume method (FVM):
The equation is:
$$
\frac{\mathrm{d} \textbf{T}}{\mathrm{d} t} = \frac{\partial \...
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0
votes
1
answer
191
views
Finite volume method for 1D heat equation in 1D
I wish to solve the following using the finite volume method:
$$\frac{\partial u}{\partial t}=\frac{D}{r}\frac{\partial}{\partial r}\left(r\frac{\partial u}{\partial r}\right)+Q(t,r)$$
with the ...
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votes
1
answer
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How can I correctly determine velocity of a point inside a grid after using mixed finite element method to solve Poisson equation?
I am using the mixed finite element method (MFEM) to solve the Poisson equation:
$$\Delta h = 0,$$where $h$ denotes hydraulic pressure. The MFEM could determine the normal flux rate, $q_n$, through ...
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votes
2
answers
81
views
Setting up consistency tests in FVM
I am in need of some help to valdate a consistency test with a finite volume method solver.
The idea is the following:
Based on the method of manufactured solutions (MMS) I am supplying the analytical ...
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2
votes
1
answer
138
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Solving a set of mixed conservative/non-conservative equations with the finite volume method
I want to solve this set of 2D advection-diffusion equations of this form in spherical coordinates:
$$
\frac{\partial f}{\partial t}=-\mathbf{u}\cdot\nabla f+\eta\nabla^2f+\eta_1(\mathbf{e}_1\cdot\...
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0
answers
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TVD slope / flux limiters formulation
Even if the formulation is the same the TVD slope limiter can be applied:
to state reconstruction at the interface, in 1D FV formulation, we reconstruct the $Q^*_{j+1/2}$ and the $Q^*_{j-1/2}$ in the ...
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votes
1
answer
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The effect of grid size on the total flux when solving Darcy flow with mixed finite element method
I am solving Darcy flow now with mixed finite element method. The Dary flow is
$$\begin{equation}\begin{aligned}k^{-1}\mathbf{q} + \nabla h=0, \text{ in } \Omega\\ %
\nabla\cdot \mathbf{q} = 0, \text{ ...
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How to assign initial velocity field and handle pressure-velocity coupling in FVM?
I am trying to solve the 2D incompressible Navier-Stokes equations for laminar flow over a backward facing step using the finite volume method.
This is the plot that I generated of a generic mesh ...
0
votes
1
answer
170
views
Implementation of mixed hybrid finite element method
The mixed hybrid finite element method (MHFEM) is based on the mixed finite element method (MFEM). So, I'd recall the implementation of MFEM.
The mixed formulation of Poisson equation reads
$$\begin{...
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votes
1
answer
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Finite volume method on a nonuniform grid
I would like to ask a question on the implementation of finite volume method on a non-uniform grid in solving Navier-Stokeq equations. I will just post the screenshot of a PhD thesis, where I found ...
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1
answer
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Elementary matrix of Raviart-Thomas elements
We can use the $RT0$ to solve the Darcy equation, i.e.
$$k^{-1}\mathbf{u}+\nabla p = 0, \text{ in } \Omega,$$
$$-\nabla \cdot \mathbf{u} = 0, \text{ in } \Omega,$$
$$p = p_D \text{ on } \partial\Omega,...
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votes
2
answers
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Test functions of Raviart-Thomas elements?
The test functions of general finite elements are like interpolation functions (if my understanding is correct). But how about test functions of Raviart-Thomas elements?
Let's raise the $RT0$ element ...
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votes
2
answers
170
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Numerical methods for Vlasov's equation
Vlasov equation is pretty straightforward
It would be easy to solve with Fem packages like firedrake, but in my case I have 6d distribution function: it depends on 3d vector of spatial coordinates ...
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votes
1
answer
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What determines the order of a finite volume scheme?
I often hear that cell centred finite volume is second order accurate but at the same time I come across notions of high order FVM flux schemes. Is there a distinction between the two? If I were to ...
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2
answers
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Why aren't face integrals for an element calculated in FEM but they show up in FVM?
Consider the Laplace problem:
\begin{align}
-\nabla^2 u = f \qquad \text{in } \Omega \\
u = 0 \qquad \text{on } \Gamma
\end{align}
The weak problem is find $u_h \in V \subset H^1$ such that $\...
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votes
1
answer
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views
Discretizing the viscous component in 1 - D Navier stokes compressive flow
I've been working on modelling the NS equations in order to simulate shock waves. The equations are set up on the form:
\begin{equation}
\frac{\partial U}{\partial t} + \frac{\partial F(U)}{\partial x}...
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1
vote
1
answer
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QUICK scheme derivation
I am reading about QUICK scheme for calculating the value of unknown variable $\phi$ in finite volume method. Given a locally one dimensional flow, we assume the value of $\phi$ is computed as a 2nd ...
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0
answers
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How to determine the orientation of convex/concave hexahedra?
I am writing a code that checks the orientation of a list of vertices (along with face connectivity) describing both convex and concave hexahedra. The face connectivity table stores the list of vertex ...
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votes
1
answer
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Finite Volume on Cubed Sphere
The US weather model uses an uncommon (?) discretization called 'Finite Volume on Cubed Sphere'.
To avoid the singularities that occur at the poles when using lat/lon discretization, they instead ...
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Discretizing Multi-species Ion Exchange Equations by Finite Volume Method
I'm solving a system of multispecies ion exchange equations (diffusion+drift fluxes) in 1-d spherical domain using finite volume method to obtain the ion concentrations at the next time step. After ...
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2
answers
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What is the rationale of second-order finite volume discretization?
When it comes to a second-order accurate finite volume discretization of Navier-Stokes equations, which one of the two following rationales is adopted?
1- Second-order accuracy is a direct consequence ...
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votes
2
answers
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Is mesh orthogonality important for FEM?
While studying mesh quality metrics in literature and software documentation, I've seen discussions about mesh orthogonality in Finite Volume Method (FVM) contexts, but not for Finite Element Method (...
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1
answer
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How is a wall boundary implemented (using ghost cells) in a simple 2d euler flow solver?
I understand that you must reflect the velocity of the cell across the wall and store that reflected velocity in the ghost cell (which will then be used for flux/residual calculations), but that is ...
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1
answer
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why is the third flux term in the conservative euler equation positive regardless of the direction of velocity?
The $\rho u^2 + p$ term corresponds to the flux of $x$-momentum through some surface. According to this equation it would give a positive momentum flux if $u$ was positive or negative. This does not ...
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1
answer
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How are eigenvalues of a cell in the finite volume discretization calculated? (for euler fluid equations)
I am building a program to numerically solve the euler fluid equations based on finite volume discretization and am having trouble on a step where the eigenvalues of a cell need to be calculated ...
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votes
0
answers
44
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The physical meaning of conservative mass in diffusion equation
I am working on 1-D mass transient diffusion in a radial domain (spherical object) using finite volume method. My equation reads
$$\frac{\partial C}{\partial t} = D\left[\frac{\partial^2 C}{\partial r^...
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6
votes
1
answer
244
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Projection method FVM poisson part, adding source term
The idea of the method is to decompose the Navier-Stokes equation into the solenoidal and irrotational parts.
$$\frac{\partial u}{\partial t}+u(\nabla \cdot u)=-\frac{1}{\rho}\nabla p+\nabla ^2 u$$
...
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votes
1
answer
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Discretization of a non-linear ODE using FDM isn't grid indepenent
I am trying to solve the ODE :
$\frac{d^2T}{dx^2} = \omega_1 T+\omega_2 T^2$
+
using different numerical methods. I have tried the following discretizations so far and none of them seem to be grid ...
0
votes
1
answer
234
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Unstructured mesh preprocessing
For solving PDE with self written code it is needed to preprocess the data from mesh generators. I recently started shifting from cartesian grid to unstructured.
I finished reading up to FVM part of ...
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