Questions tagged [finite-volume]

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

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45 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 ...
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110 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 $\...
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104 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 ...
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1answer
71 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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27 views

shifting mass along a vector field

I have a positive matrix $\rho \in \mathbb{R}^{n,n}_+$ as a discrete probability density. Furthermore, I have a tensor $u \in \mathbb{R}^{n,n,2}$ that is constant in time and acts as a vector field. I ...
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1answer
660 views

How to make a less diffusive code to solve 2D advection equation?

I would like to solve the following differential equation numerically in 2D, $$\frac{\partial z^-}{\partial t}+(\vec{B}\cdot\vec{\nabla})z^-=0,$$ see Wikipedia if you are curious about what the ...
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1answer
53 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 ...
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52 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 ...
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52 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 ...
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1answer
67 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 ...
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52 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 ...
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2answers
320 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 (...
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1answer
55 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 ...
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1answer
62 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 ...
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1answer
123 views

How do I find the portion of a cell/voxel lying within a defined surface?

We have a 3-dimensional grid of voxels (or cells), with individual voxels being of volume $dx\,dy\,dz$ where $dx=dy=dz=1$. A cone-like surface is defined by some function, $z = f(x, y)$, which in ...
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161 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. $$\frac{\partial u}{\partial t}+u(\nabla \cdot u)=-\frac{1}{\rho}\nabla p+\nabla ^2 u$$ ...
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1answer
271 views

Elliptic PDE finite volume method with Dirichlet boundary condition

I want to discretize the following equation using a Finite Volume Method $$\nabla \cdot (a(x)\nabla u)=f(x)\\x\in \Omega \subset \mathbb{R}^2 \\u_{|\partial\Omega}=g$$ I'm using Voronoi cells here: ...
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41 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^...
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1answer
113 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 ...
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1answer
106 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 ...
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1answer
77 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 ...
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92 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 ...
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42 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 ...
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1answer
68 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, ...
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2answers
227 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{...
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2answers
198 views

Finite Volume Method flux integration

I was reading through this document about FVM. I understood all up to the point where we have on page 15 the following $$(\bar u^{n+1}_i - \bar u^n_i) \Delta x + \int^{t^{n+1}}_{t^n} f(u(x_{i+1/2},t))...
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1answer
41 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 ...
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1answer
63 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 ...
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61 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 ...
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1answer
132 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 ...
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2answers
135 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 ...
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36 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 ...
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1answer
161 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 ...
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1answer
583 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 ...
2
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3answers
240 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 ...
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1answer
107 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 ...
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3answers
326 views

Is there a minimum angle requirement for cells in the finite volume method?

In his talk "What is a good linear finite element?", Shewchuk states that small dihedral angles in linear tetrahedra elements cause ill-conditioning of the stiffness matrix. Do small dihedral angles ...
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2answers
185 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 ...
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0answers
41 views

Cell-centered DG extension to the two-point flux approximation scheme

A current problem that I am working on requires me to compute the solution from the heat diffusion evolution on a discontinuous function. More precisely - I have a Delaunay triangulation and within ...
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0answers
54 views

Solving Stokes Equations in 3D - Do I need to treat pressure-velocity coupling iteratively?

I need to develop a code to solve Stokes Equations in 3D in cubic geometries (structured grid, uniform mesh spacing). My code needs to take a pressure gradient in one direction as a BC (pinlet=p1, ...
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0answers
41 views

Effect of reducing flux consistency order at boundary on convergence order

Consider the 1D nonstationary convection-diffusion PDE $$ \begin{alignat}{2} \partial_t u &= -a \partial_x u + D \partial_{xx}u, &\qquad x \in (0,1), t \in (0,T), \\ f(t) &= \left.\left( a ...
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156 views

Two-dimensional modelling of a flate-plate reactor

I am trying to simulate the unsteady pollutant concentration along the reactor by solving the 2nd order PDE below with the stated BC and IC. which method is appropriate to solve convection-diffusion-...
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1answer
229 views

Without positive definiteness, does an iterative solver work?

Question Does lacking positive definiteness of the matrix of coefficients in a system of equations, make using iterative solvers impractical? Description Using the finite volume method, I have ...
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46 views

Discretisation by FVM with triangulare mesh

I work on electromagnetic modeling, and I use the finite volume method as a tool, so I want to develop code by the modified finite volume method (with triangular mesh), my problem studied is in a ...
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8answers
6k views

What programming language should I choose and why?

I am a mechanical engineer, intermediated/advanced level in MATLAB and MATHEMATICA, and beginner in Python. I intend to get a PhD in aeroelasticity (FEM + CFD) and coding my own program. I intend to ...
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0answers
24 views

Convection equation expanded in Legendre polynomials

In some areas of physics, for example radiation transport in plasmas, it is beneficial to re-express the convection equation in terms of a Legendre polynomial expansion. This results in a set of ...
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0answers
66 views

How does the "Stable Fluids" algorithm by Jos Stam relate to the SIMPLE and PISO algorithms?

The "Stable Fluids" paper (*) by Jos Stam starts by acknowledging that "Our method cannot be found in the computational fluids literature, since it is custom made for computer graphics applications. ...
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66 views

The relation between PDE order and discretization order

In Jasak's Ph.D. thesis (2000), a notion is given about discretization of a transport equation: For good accuracy, it is necessary for the order of the discretization to be equal to or higher than the ...
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3answers
343 views

How is central difference scheme second-order accurate?

In an arbitrarily unstructured mesh, shown in the figure below, in the context of finite volume method, I want to obtain an approximation of $\phi_f$, where $N$ and $P$ are cell centers of adjacent ...
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2answers
275 views

Numerical flux and source term in FVM (Burger's like equation)

I'm trying to solve the following equation with FVM $$u_t + f(u)_x = g(u)$$ where $g$ is some smooth function of $u$ and $f(u) = \frac{u^2}{2}$. This is really similar to Burger's equation, except ...

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