Questions tagged [discretization]

The process of representing a continuum space with a finite set of points/elements

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331 views

Global truncation error behavior at fixed time step

I am trying to solve the following diffusion equation problem: $\frac{\partial f}{\partial t}=\frac{\partial (D\frac{\partial f}{\partial x})}{\partial x}+S$ $D=1+x^{2}+\sin(x)$ $f(x,0)=1 , f(0,t)...
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3answers
123 views

Discretization Error amplification instead of stagnation to machine precision

I wrote a code on Python 2.7.5 to solve numerically the following differential equation. $\frac{\partial^2f}{\partial x^2}=-S$ $S=\pi^{2}\sin(\pi x)$ S is chosen that way in order to have $f= \sin(\...
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1answer
56 views

Problems with deriving an equation for a finite-difference scheme given in the journal paper

I'm reading this paper and trying to follow everything that the author has done. A Comparative Study of Finite Volume Method and Finite Difference Method for Convection-Diffusion Problem But there ...
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118 views

Monotone, monotonicity preserving, LED, TVD, NVD, bounded, stable and stability preserving discretization schemes [closed]

When it comes to discretization schemes for finite volume method, the following terms can be found in literature: monotone schemes monotonicity preserving schemes local extremum diminishing schemes ...
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5answers
903 views

Don't we care about the numerical diffusion in the diffusion term?

In the context of the solution of advection-diffusion equations by finite volume method, many numerical schemes, papers and book chapters are dedicated to address the numerical diffusion and/or ...
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109 views

How to account for the interface between two different phases in a discretized diffusion model?

I have tried to set up a model for the diffusion of a gas into a liquid. The two media are next to each other and the geometry is spherical because the system should simulate the diffusion out of a ...
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1answer
151 views

Discontinuous Galerkin FEM : Control points are mid-points of edges instead of nodes

I am thinking to use discontinuous galerkin FEM (DGFEM) method to estimate discontinuous displacement field $u: \Omega \rightarrow \mathbb{R}^2$ at the crack surface of a material. The domain is ...
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1answer
46 views

Implementation of stochastic cellular automata

In my problem, I have a lattice with a stochastic cellular automaton. In order to simplify a bit, let's say it is 1D. In my system, each node can be type A, B or C. A way to represent the system and ...
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1answer
162 views

Can this nonlinear advection-diffusion equation be discretized as to only have to solve SPD systems?

Consider a nonlinear advection-diffusion equation of the form $$ \frac{\partial u}{\partial t} = \nabla \cdot (a(u) \nabla b(u) - \vec{c}(u)u) \tag{1} $$ on a rectangular domain with Dirichlet ...
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385 views

Implement Robin boundary condition (finite volume)

I have a PDE equation with Robin Boundary condition in an annulus system and I should solve it by finite volume method: \begin{align} \frac{\partial T_f}{\partial t} - k \left(\frac{\...
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82 views

Discrete operator textbooks

This will be a vague question. When I was writing a finite element matrix assembly routine, a colleague noticed that I had a bug in my code because the sparsity pattern of the one of the blocks didn'...
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512 views

Discrete wave simulation - absorbing boundaries?

I wrote a simple 2D wave simulation using the following equations: $$\frac{\partial^2 u}{\partial t^2}=c^2\nabla^2u$$ Where $\nabla^2$ is the discrete laplace operator using a Von Neumann neighborhood ...
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1answer
358 views

Finite Volume Polar Discretization: Lengths

Given a uniform polar grid, as in the figure below: and a FV discretization of a gradient for example: $\frac{\partial p}{\partial \varphi} = 0$ $\Delta r \frac{p_e - p_w}{\Delta \varphi} = 0$ My ...
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81 views

Literature on numerical solving based on multiple meshes?

Consider solving a differential equation system for 1D, 2D, or 3D. It involves various input and output "field" variables, which, more often than not, correspond to various physical quantities ...
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548 views

Order of accuracy of FVM discretization

I've recently got interested in CFD and started a small project by solving the radial Reynoldsequation. Why the Reynoldsequation? I recently encountered it through my studies and somehow got stuck :) ...
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447 views

Rhie and Chow Pressure Velocity Coupling

In a collocated grid, which velocity is used in the convective term in the momentum equation? Is the Rhie and Chow constructed face velocity or an average of the adjacent cell center values(in a ...
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79 views

Three steps of pde numerical solution and nonlinear equation

I'm very new here. I'm trying to solve nonlinear elliptic equation $$ (n(u)u')' = f(u) $$ and face with crucial misunderstanding. As I suppose, the procedure of solving some nonlinear equation ...
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1answer
209 views

Use Finite Difference Discretization to find approximate solution to the Poisson's equation

I've just been introduced to the Poisson's equation. I've never had the need to dealt with PDE, so I'm a bit lost. Apparently we can compute an approximate solution of the Poisson's equation $$\frac{\...
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2answers
190 views

Combining trapezoidal rule with upwind scheme

I want to discretize and numerically solve the equation: \begin{equation} v(k)\dfrac{\partial f}{\partial z} + F(z)\dfrac{\partial f}{\partial k} = \alpha f(z,k) \,\,. \end{equation} Discretization ...
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1answer
396 views

Mass Lumping in case of Dirichlet boundary conditions

I'm currently trying to implement a FEM to solve a type of wave equation with homogeneous Dirichlet boundary conditions by using standard $\mathcal{P}_1$ triangle elements and an explicit scheme for ...
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0answers
37 views

Inner Products vs Discretizations of Functions

Let $f:(0,1)\rightarrow [0,1]$ be a $\mathcal{C}^{\infty}$ function. We say that a function $D_r^f:\{1,...,r\}\rightarrow [0,1]$ is an $r$-discretization of $f$ if $$D_r^f(j) = \frac{1}{|a(j)-b(j)|}\...
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1answer
137 views

Space-time Galerkin of Burgers changes the convection speed

tldr: Can space-time Galerkin schemes applied to convection-diffusion problems lead to effects on the convection velocity? For time $t\in (0,1)$ and the spatial variable $\xi \in (0,1)$, I am ...
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2answers
273 views

Motivation behind Collocation Method

In the previous question "Motivation behind Galerkin method", Paul gives a good and easy-to-understand explanation indicating that the Galerkin method is a kind of projection method. Can anyone ...
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55 views

Second and Higher Order Order Corrector in Spectral Deferred Correction

I am trying to work out a second order or higher order correction for the method of Spectral deferred Correction (SDC). Specifically using as a corrector a second order or third order multi-step. In ...
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1answer
147 views

High order unconditionally stable discretization for a scalar hyperbolic PDE

In order to numerically solve the following differential equation: \begin{equation} \text{Fr}\{f\} := v(k)\dfrac{\partial f(z,k)}{\partial z} - F(z) \dfrac{\partial f(z,k)}{\partial k} = -\dfrac{f-...
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1answer
741 views

Second order interpolation scheme

On a grid I am having the values of a physical quantity say for example Temperature, at the E,W,N,S and P node all of them being calculated using a second order discretization scheme. I want a second ...
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1answer
466 views

How to define residual in multigrid approach?

I wish to solve the two-dimensional Navier Stokes equations using multigrid method on a collocated grid using the Predictor-Corrector method mentioned below. But first, let me elaborate on what I had ...
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2answers
50 views

How to choose a good distribution for visualizing phase changes in the nature of the roots of a quadratic equation

I'm not sure if this SE site is the best one for this question, so let me know where it should be moved to if you think it doesn't belong here. After learning about the quadratic formula, I'm ...
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70 views

Does scaling factor affect discretization?

Suppose I want to solve the below equation numerically. $$ \frac{dy}{dx}=y $$ I'd like to normalize the space discretization by choosing $$ a\bar{x}=x $$ where I assume $\bar{x}$ is unity. Then the ...
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1answer
168 views

Apply second order finite difference discretization for mixed boundary condition

I want to solve the problem below \begin{equation} \begin{aligned} \eta u-\Delta u &=f, &\text{in $\Omega$}\\ \end{aligned} \end{equation} where $\Omega=(0,1)\times (...
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1answer
555 views

Discretization method for advection equation without numerical diffusion

Given the advection equation for an incompressible flow field $$\frac{\partial c}{\partial t} + \mathrm{Pe} \frac{\partial c}{\partial x} = 0$$ what would the best method be for discretizing this ...
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1answer
148 views

Boltzmann equation and equilibrium distribution function

The free-streaming term in stationary 1D Boltzmann equation satisfies the equilibrium distribution function, that is: \begin{equation} \text{Fr}\{f\} := v(k)\dfrac{\partial f(x,k)}{\partial x} - \...
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41 views

Stability in Discretization of 1D Stationary Boltzmann equation

I want to discretize and numerically solve the following PDE: \begin{equation} v(k)\dfrac{\partial f}{\partial x} + E(x)\dfrac{\partial f}{\partial k} = S\{f\} \end{equation} using finite volume (box ...
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1answer
749 views

Finite difference recursion and higher order

This may be a trivial question, but I've always wondered... For the classical, central finite difference schemes, if I'm interested in determining the second derivative, does applying the first ...
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1answer
248 views

Stability in discretization of a PDE

Suppose I want to numerically solve for $f(x,k)$ the one-dimensional Boltzmann equation for electrons in steady-state condition, given as: \begin{equation} \left( \dfrac{\hbar k}{m} \right) \dfrac{\...
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1answer
394 views

a few questions on understanding geometric conservation law

I was wondering if anyone could help with understanding the geometric conservation law for moving domains. I came across Link1, and have tried to understood the paper by Farhat et al Link2. So far ...
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3answers
254 views

solving PDEs in MATLAB

I want to solve 3 coupled equations. I converted them to a system of odes in time and discrete it in Length and radius. Now I have a problem in one of the equations in first point. Because in this ...
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1answer
2k views

PDE - Conservative form, conservative methods and discrete conservation

I cannot find a reference explaining clearly and rigorously the links between the notions of conservative form for a PDE, a conservative numerical method and discrete conservation. I would be very ...
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47 views

Order of accuracy of linearised vs non-linear system

Does the order of accuracy of a combination of schemes applied to solve a system of non-linear equations, match those of the same schemes applied to the linearised version of the system? In other ...
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1answer
4k views

what is the difference between non-conformal and conformal?

So far what I understand is that two neighbouring elements are conformal if their edges and faces match exactly, whereas with non-conformal elements this is not the case. For instance, h-refinement ...
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2answers
187 views

Is it possible to show global conservative properties FEM as it is done in FVM?

I know that in FVM, it is possible to show that a discretisation scheme is conservative by adding the discrete terms over a few control volumes and showing that all terms cancel apart from those ...
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189 views

Do collocated grid arrangements definitely result in the checkerboard effect?

I understand the checkerboard effect due to the use of collocated grid arrangements in FVM. However, I wanted to know whether this problem is definitely bound to effect the results? For instance, I ...
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1answer
230 views

Are FEM or DGFEM methods based on integrals or PDEs?

I know that FVM is based on the integral form of conservation laws, and FDS is based on PDEs. What I'm confused by, is whether FEM and DGFEM formulations are based on integral or pde form of ...
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0answers
92 views

problem with understanding the fluid boundary conditions of a 1D probelm

I am having problems understanding the boundary conditions of the problem described in this paper on researchgate Essentially the problem consists of a one dimensional fluid chamber in contact with a ...
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2answers
267 views

How to separate text from the paper on a black and white page?

I tried to discretize an image into black and white and came into some difficult. The difference between the letters and paper is pretty clear to our eyes: However a simple thresholding trick doesn'...
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0answers
589 views

Time discretization of wave equation

I am trying to model the seismic wave equation and have therefore been reading about discretization schemes and their stability. I recently came across an insightful paper on 'Galerkin FEM methods for ...
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1answer
271 views

How to discretize this integral equation? (Langevin Eq)

I am trying to build my own simulator of Langevin Equation for the Brownian motion. According to this material. The way we calculate the particle position in certain time step is : W(u) is a Wiener ...
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0answers
120 views

Finite difference aproximation - Darcy law

I am solving following problem: Filtration of water can be described in bi-dimensional case by $$- \partial_x(K(x,y) \partial_x u ) - \partial_y (K(x,y) \partial_y u ) = 0, $$ where $u$ - water ...
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1answer
90 views

solving a hyperbolic set of equations - upwind type method

I want to solve a set of hyperbolic equations (not the Euler equations) using an upwind type method. I am interested in using a first order upwind scheme and one that is not based on the method of ...
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2answers
178 views

Why have specialised upwind schemes been developed to solve hyperbolic equations?

Are upwind schemes such as Godunov type methods superior to central differencing schemes? Do the reasons include superiority in modelling hyperbolic problems with Dirichlet BC's?