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Questions tagged [finite-volume]

Referring to the discretization of partial differential equations using Finite Volume Method.

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Lowest order Raviart Thomas elements

I have questions regarding the implementation of Lowest order Raviart Thomas elements on quadrilaterals, some papers use the basis functions in this form: for example the right edge: $N = \left[\dfrac{...
Amr Ashraf Ibrahim Ibrahim's user avatar
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Finding neighboring cells using Gmsh API

I am creating a simple mesh on a square domain, $[-5,5]\times[-5,5]$ using the following .geo file ...
Mainak's user avatar
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Need help with adaptive meshing code

I am trying to understand adaptive meshing and is using this code (https://github.com/esquivas/amr1d) as a reference. However, there is no documentation for it and thus, it is hard for me to ...
newbie125's user avatar
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numerical schemes for 1D PDE: for smaller grid size there is an increased roundoff error, larger size more truncation, so sweet spot in between?

I had a discussion with a colleague today. He claimed that usually for a general numerical scheme for solving a general 1D PDE, for smaller grid size there is an increased roundoff error because of ...
Millemila's user avatar
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1 answer
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Generating unstructured finite volume mesh

I want to generate a triangular mesh over a rectangle domain in order to solve Euler equations. Most mesh generator generate a mesh while providing node connectivity for each element. This is ...
L Maxime's user avatar
2 votes
1 answer
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Creating nonuniform grids for FDM with multiple points of concentration

If I am creating a grid in the $S_i$ direction with $N_S+1$ grid points. If I want more steps around some $K$, I can use: $$ S_i=K+c \sinh \left(\xi_i\right), \quad i=0,1, \ldots, N_S $$ where $c=\...
THATS MY QUANT MY QUANTITATIVE's user avatar
1 vote
0 answers
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Finite volume method for a general flux

How to approximate flux 𝐹(𝑢)⋅𝑛 where 𝑛 denotes the unit normal outward when using finite volumes? in my case it's not a conservation law so my question is how can we approximate the final term \...
Chems Eddine's user avatar
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1 answer
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How to approximate the flux when using finite volumes?

How to approximate flux $F(u)\cdot n$ where $n$ denotes the unit normal outward when using finite volumes? $$\int_{\sigma} F(u) \cdot \boldsymbol{n}_{K, \sigma} \mathrm{d} \gamma(x)$$
Chems Eddine's user avatar
3 votes
1 answer
154 views

Non-standard boundary condition for incompressible Navier Stokes

I am having difficulties applying the boundary condition $$\frac{\partial \vec{V}}{\partial t} + u\frac{\partial \vec{V}}{\partial x} = \frac{1}{\operatorname{Re}}\frac{\partial ^2 \vec{V}}{\partial y^...
2Napasa's user avatar
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4 votes
1 answer
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Burger's equation (PDE) does not work with downwind difference?

I'm working on implementing the discretised Burger's equation. I am quite confused as to why it does not work when using a step-function and downwind difference formula. When using a step-function and ...
blov's user avatar
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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{...
VIVID's user avatar
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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{\...
user46777's user avatar
2 votes
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}),(...
Matthew Hunt's user avatar
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2 answers
138 views

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?
L Maxime's user avatar
2 votes
1 answer
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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 ...
2Napasa's user avatar
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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 ...
Kbzon's user avatar
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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. ...
Diksha's user avatar
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3 votes
2 answers
201 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 +\...
BitterDecoction's user avatar
1 vote
1 answer
174 views

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 ...
Dan Doe's user avatar
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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 ...
Xray25's user avatar
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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 ...
Sayan's user avatar
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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 ...
Sayan's user avatar
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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, ...
Mainak's user avatar
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0 answers
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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 \...
Dan Doe's user avatar
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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 ...
Chi Chi 's user avatar
1 vote
1 answer
91 views

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., ($\sin(s)\;cos(s)$). ...
mysn's user avatar
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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 ...
nicholaswogan's user avatar
1 vote
1 answer
66 views

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 \...
user avatar
0 votes
1 answer
218 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 ...
Matthew Hunt's user avatar
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1 answer
77 views

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 ...
Tingchang Yin's user avatar
3 votes
2 answers
86 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 ...
Banana trick's user avatar
2 votes
1 answer
144 views

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\...
BitterDecoction's user avatar
1 vote
0 answers
84 views

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 ...
albiremo's user avatar
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2 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{ ...
Tingchang Yin's user avatar
2 votes
0 answers
75 views

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 ...
4th_ord_Padme_scheme's user avatar
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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{...
Tingchang Yin's user avatar
2 votes
1 answer
534 views

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 ...
jengmge's user avatar
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2 votes
1 answer
200 views

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,...
Tingchang Yin's user avatar
3 votes
2 answers
400 views

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 ...
Tingchang Yin's user avatar
0 votes
2 answers
188 views

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 ...
Moonwalker's user avatar
4 votes
1 answer
521 views

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 ...
CuteCompute's user avatar
1 vote
2 answers
288 views

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 $\...
CuteCompute's user avatar
3 votes
1 answer
117 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}...
Twm1995's user avatar
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1 vote
1 answer
308 views

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 ...
Algo's user avatar
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1 vote
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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 ...
niran90's user avatar
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3 votes
1 answer
249 views

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 ...
guest's user avatar
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0 answers
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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 ...
Matt's user avatar
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1 vote
2 answers
373 views

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 ...
Naghi's user avatar
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3 votes
2 answers
717 views

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 (...
Eduardo's user avatar
  • 141
0 votes
1 answer
270 views

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 ...
Frosty's user avatar
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