Questions tagged [finite-volume]
Referring to the discretization of partial differential equations using Finite Volume Method.
257
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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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0
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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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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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1
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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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73
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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
$$\partial_t \boldsymbol u(t,\boldsymbol x) + \nabla \cdot \...
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0
answers
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Navier Stokes Equation Discretization by upwind scheme
I've been reading this paper regarding the SIMPLER algorithm in CFD and I don't understand exactly how the discretized equation mentioned by the author is arrived at. It says that the convective terms ...
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Parallel Plate Flow simulation using SIMPLER Alogortithm
I am new to the community and CFD itself and am working on my first CFD program in Scilab based on the SIMPLER algorithm in Scilab i.e. modeling a flow between parallel plates with entrance region. I ...
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Can I use MOL to solve 2D steady state PDE in terms of r and z spatial coordinates?
recently I need to solve a 2D steady state PDE equation.
It’s not time dependent, and the only two independent variables are z and r direction.
So far for this solution, I was thinking using Method ...
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1
answer
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How to solve advective equation with source term depending on variable
I have the following equation
$$
\dfrac{\partial s}{\partial t} + \nabla \cdot \left( \vec{v} s\right) = f(s)
$$
Where $f(s)$ is an explicit source term that depends on $s$, e.g., ($\sin(s)\;cos(s)$).
...
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Why does the non-orthogonal term in FVM need to be evaluated explicitly?
In FVM, the face normal gradient is split into orthogonal and non-orthogonal part. The orthogonal part is evaluated implicitly, whereas the non-orthogonal part is evaluated explicitly as a source term ...
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Centered finite volume scheme for an advective term on an unstructured/irregular/non-uniform grid
Consider the continuity equation
$$\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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votes
0
answers
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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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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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2
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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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votes
1
answer
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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:
$$
\frac{\partial f}{\partial t}=-\mathbf{u}\cdot\nabla f+\eta\nabla^2f+\eta_1(\mathbf{e}_1\cdot\...
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0
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TVD slope / flux limiters formulation
Even if the formulation is the same the TVD slope limiter can be applied:
to state reconstruction at the interface, in 1D FV formulation, we reconstruct the $Q^*_{j+1/2}$ and the $Q^*_{j-1/2}$ in the ...
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1
answer
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The effect of grid size on the total flux when solving Darcy flow with mixed finite element method
I am solving Darcy flow now with mixed finite element method. The Dary flow is
$$\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
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1
answer
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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
$$\begin{...
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1
answer
257
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Finite volume method on a nonuniform grid
I would like to ask a question on the implementation of finite volume method on a non-uniform grid in solving Navier-Stokeq equations. I will just post the screenshot of a PhD thesis, where I found ...
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1
answer
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Elementary matrix of Raviart-Thomas elements
We can use the $RT0$ to solve the Darcy equation, i.e.
$$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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2
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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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2
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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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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
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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
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answer
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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
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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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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
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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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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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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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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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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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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
$$\frac{\partial C}{\partial t} = D\left[\frac{\partial^2 C}{\partial r^...
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1
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Projection method FVM poisson part, adding source term
The idea of the method is to decompose the Navier-Stokes equation into the solenoidal and irrotational parts.
$$\frac{\partial u}{\partial t}+u(\nabla \cdot u)=-\frac{1}{\rho}\nabla p+\nabla ^2 u$$
...
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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
165
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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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votes
1
answer
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FV Discretization of source term in 2D Poisson Equation
I am learning Finite Volume method using the textbook "An Introduction to CFD: Finite Volume Method" by Veersteg and Malalasekera.
I would like to solve a heat conduction problem over a ...
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0
answers
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How are finite volume method boundary conditions implemented without using ghost-cells?
I'm currently trying to implement my own FVM code in cpp, but when I try to calculate the laplacian of a test function, given by
\begin{align}\phi_0=\sin(2\pi x)\sin(2\pi y),\end{align}
I get ...
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Are FDS scheme and ROE-FDS scheme the same?
In SU2 documentation about the numerical schemes (https://su2code.github.io/docs_v7/Convective-Schemes/) FDS is mentioned as a "standalone" method.
Yet, when looking online for more ...
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answer
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How does the diffusion of a finite volume method with a WENO scheme compare with that of spectral methods?
I know that, in general, finite volume (FV) methods are more (numerically) diffusive than spectral methods. However, I can't find any information on how the advection scheme changes that.
For example, ...
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2
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How to apply central difference to viscous fluxes in 2D Navier-Stokes equations?
I'm trying to solve 2D unsteady compressible Navier-Stokes equations with finite-difference or finite-volume method. Here is the system, it's pretty standard:
$$
\frac{\partial U}{\partial t} +
\frac{...
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1
answer
49
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How to evaluate the average value of a polynomial inside the triangle area in finite volume sense?
Consider we have a linear bivariate polynomial:
$$p(x,y)=ax+by+c.$$
To construct the linear polynomial using least square method, we need to evaluate the value of the average polynomial $p$ in at ...
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vote
1
answer
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Where could error terms that blow up in SWE come from?
I have been working on a solver for shallow water equations with reflective boundary conditions. I have found that it diverges very fast. As a workaround I noticed yesterday that if I smooth the ...
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Finite volume reconstruction techniques
For a cell-centered finite-volume calculation, with a nontrivial stencil (either using an unstructured grid or a strongly non-uniform structured grid), what are the main techniques for reconstruction ...
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