Questions tagged [finite-volume]

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

Filter by
Sorted by
Tagged with
0 votes
0 answers
19 views

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 ...
0 votes
0 answers
49 views

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 ...
1 vote
1 answer
55 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., ($\sin(s)\;cos(s)$). ...
0 votes
0 answers
47 views

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 ...
2 votes
0 answers
46 views

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 ...
1 vote
1 answer
62 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 \...
0 votes
0 answers
112 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 ...
0 votes
1 answer
58 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 ...
4 votes
2 answers
74 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 ...
2 votes
1 answer
111 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\...
1 vote
0 answers
49 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 ...
  • 410
2 votes
1 answer
66 views

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{ ...
2 votes
0 answers
64 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 ...
0 votes
1 answer
122 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{...
2 votes
1 answer
193 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 ...
  • 51
2 votes
1 answer
141 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,...
3 votes
2 answers
244 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 ...
0 votes
2 answers
115 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 ...
4 votes
1 answer
156 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 ...
1 vote
2 answers
248 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 $\...
3 votes
1 answer
102 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}...
  • 55
1 vote
1 answer
187 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 ...
  • 294
1 vote
0 answers
67 views

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 ...
  • 213
4 votes
1 answer
130 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 ...
  • 43
0 votes
0 answers
56 views

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 ...
  • 1
1 vote
2 answers
211 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 ...
  • 235
3 votes
2 answers
466 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 (...
  • 141
0 votes
1 answer
170 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 ...
  • 27
0 votes
1 answer
66 views

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 ...
  • 27
0 votes
1 answer
91 views

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 ...
  • 27
0 votes
0 answers
42 views

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^...
  • 1
7 votes
1 answer
217 views

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. $$\frac{\partial u}{\partial t}+u(\nabla \cdot u)=-\frac{1}{\rho}\nabla p+\nabla ^2 u$$ ...
  • 354
0 votes
1 answer
154 views

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
145 views

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 ...
  • 354
2 votes
1 answer
187 views

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 ...
  • 145
2 votes
0 answers
284 views

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 ...
  • 31
1 vote
0 answers
60 views

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 ...
  • 11
3 votes
1 answer
89 views

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, ...
  • 155
1 vote
2 answers
373 views

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{...
  • 347
1 vote
1 answer
47 views

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 ...
1 vote
1 answer
68 views

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 ...
  • 159
1 vote
0 answers
67 views

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 ...
3 votes
1 answer
169 views

Is upwinding needed for slope limiter / flux limiter and numerical flux?

I have a cell centered cartesian grid and am trying to implement the flux inside the divergence term using numerical flux with a flux limiter. I found different formulas for MUSCL flux limiter, where ...
  • 159
2 votes
2 answers
149 views

Simplest way to "upgrade" from Euler equations to Navier-Stokes equations in FV or FD framework

I have quite a lot of experience solving unsteady Euler equations, including multi-component ones, with in house-coded finite-difference and finite-volume methods, including MacCormack and MUSCLE ...
  • 347
1 vote
0 answers
40 views

Solution of non-linear Poisson equation does not match reference

I'm trying to solve the non-linear Poisson equation as a first step to solve the drift-diffusion equations for semiconductors. For reference, I'm using a preprint from the Weierstrass Institut (which ...
  • 11
1 vote
1 answer
297 views

How to compute gradient of a cell having a boundary face?

In many situations in unstructured mesh solvers, one needs to compute gradient of arbitrary variable $\phi$ such as temperature or velocity at face centers (one of such situations is correction for ...
  • 294
2 votes
1 answer
1k views

How to calculate skewness for a mesh?

I am writing a code to calculate mesh quality stats such as: cell volume, face areas and non-orthogonality between faces (basically something like OpenFOAM's ...
  • 294
0 votes
1 answer
144 views

Is it possible to predict solution oscillation before solving the system by looking at coefficient matrix?

Question When it is about solving a system of equations, is it possible to predict that whether high-frequency noise (e.g. checker-boarding) is likely to appear in the converged solution by looking at ...
  • 235
2 votes
3 answers
350 views

Flux sign and face normal confusion in finite volume method

I implemented a solver for the 2D steady-state heat equation (without heat generation and homogeneous material) $\nabla. (k\nabla T) = 0$, using finite volume method, however, I am having some ...
  • 294
1 vote
2 answers
216 views

$P0$ elements for $H1$

Are there $P0$ (zero degree/constant element) nonconforming methods for approximating solutions in $H1$? More specifically, I have the equation: $$u-f - T\Delta u = 0$$ Which can be interpreted as ...

1
2 3 4 5