Questions tagged [finite-volume]

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

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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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116 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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98 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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Advection equation boundary conditions in 2D unstructured grid

Currently struggling with boundary conditions for 2d advection equation. left dirichlet u = (1,0) bot top dirichlet u = (0,0) right neumann du/dx = (0,0) ...
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94 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
47 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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64 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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36 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
63 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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128 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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1answer
39 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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59 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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54 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
119 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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118 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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33 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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53 views

Simple Finite Volume method for Stokes equations

I'm trying to understand how to implement fluid problems using Finite Volume Elements, for example a simple Finite Volume Elements for the Stokes problem: $-\nu\Delta u+\nabla p =f$ in a bounded ...
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1answer
117 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
297 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 ...
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1answer
103 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
177 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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171 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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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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51 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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38 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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172 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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153 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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44 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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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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151 views

Import stl file into Gmsh and create a 3d mesh for CFD

I am trying to develop a case study for teaching CFD. To this end, I am using OpenVSP, Gmsh and SU2. I have created a wing in OpenVSP, exported the CFD mesh as .stl file. Opened the .stl file in Gmsh ...
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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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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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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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231 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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88 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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244 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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62 views

2D diffusion equation using Finite Volume Method

i am working on an assignment problem: Consider a two-dimensional rectangular plate of dimension L = 1 m in the x direction and H = 2 m in the y direction. The plate material has constant thermal ...
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148 views

Mass conservation for hyperbolic relaxation problem

I have solved numerically the following system: \begin{cases} \partial_t{u} + \partial_x{v} = 0 \\ \partial_t{v} + \frac{1}{\varepsilon^2}\partial_x{u} = -\frac{1}{\varepsilon^2}(v-f(u)) \end{cases} ...
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232 views

Why FVM can handle unstructured meshes while FDM cannot?

How come Finite Volume Method(FVM) handle the unstructured meshes and Finite difference Method cannot, whereas in FVM to approximate the fluxes at the boundary we use the central differencing? My ...
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1answer
227 views

Testing a block tridiagonal system of equations

In 1D problems, tridiagonal systems of equations are obtained when we use finite-difference or finite-volumes in a structured mesh. A wide solver is the TDMA algorithm here. In two-dimensional ...
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How to avoid density getting “deleted” in two way rigid body coupling with LBM CFD?

I've been reading this paper recently, which talks about using Lattice Boltzmann methods and two way coupling. Specifically, it outlines fluid solid coupling, and solid fluid coupling, and how simply ...
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1answer
39 views

Produce vertex displacements from volumetric shrinkage data on unstructured meshes

I was wondering what would be an efficient way to produce compatible displacements for mesh nodes/vertices if the computed data is volume shrinkage of each element/cell in the unstructured mesh? ...
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2answers
164 views

FVM vs FDM vs Conservative form vs Non conservative form

My question is regarding solving the conservative form and the non-conservative form of the governing-equations (GE), like continuity or the navier stokes equation, using finite difference method (FDM)...
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162 views

CFD and finite volume method: Dirichlet boundary conditions for the Euler equations

Please point me to an answer if one already exists, but after some searching, I still can't find the answer to what seems like a very simple question. There are plenty of references out there for ...
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1answer
63 views

Obtain velocity from imposed energy spectrum using the inverse FFT

I am trying to obtain the spatial representation of $u(x)$ (e.g. velocity) from its energy spectrum $E(k)=k^4\exp(-(k/k_0)^2)$, which is given in the frequency domain, provided $|u(k)|=\sqrt{2E(k)}$. ...
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1answer
137 views

How to compute turbulent energy cascade

I need to compute the kinetic energy cascade using a finite volume solution in an equally spaced grid. I wonder if it is more correct to first compute the kinetic energy in the space (or time) domain, ...
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41 views

Grids for atmosphere simulation with finite volumes on the globe

I am currently in the early construction process of building a simple CFD model of a rotating planetary atmosphere. The planet should be allowed to tilt significantly, so that a time-dependent source ...
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114 views

Approximate $\|\Delta f\|^2_{L^2(\Omega)}$ in finite element context

I have minimization problem of the form $$ G(f) + \|\Delta f\|^2_{L^2(\Omega)} \to \min $$ over all $f\in C^2(\Omega)$, $\Omega$ being closed and bounded. Let us forgot about $G$; I'm interested in ...
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109 views

Abaqus, ANSYS, and FVM solver for thermal expansion problem converges to different values

Is it reasonable for a FEM and FVM code to converge to slightly different solutions for the same physical problem (identical BCs, geometry, properties, etc...), provided stability constraints are ...
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80 views

Computing face fluxes in FVM

In FVM, we have to compute fluxes at some face of a cell. There are many ways to compute this face flux value, but the most common and easiest way involves some simple averaging at the face. However, ...

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