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

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2
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2answers
50 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 ...
1
vote
0answers
68 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 ...
1
vote
0answers
29 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)|}\...
4
votes
1answer
86 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 ...
4
votes
1answer
89 views

Motivation behind Collocation Method

In previous question "Motivation behind Galerkin method", Paul gives a good and easy-understanding explanation indicating that the Galerkin method is a kind of projection method. Can anyone explain ...
0
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0answers
18 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 ...
4
votes
1answer
123 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-...
3
votes
1answer
102 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 ...
1
vote
1answer
130 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 ...
0
votes
2answers
45 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 ...
0
votes
0answers
32 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 ...
0
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0answers
29 views

Discretization of NORMALIZED Poisson Equation, 1-D Semiconductor

Poisson Equation $$ \frac{\partial }{\partial x}\left(\epsilon\frac{\partial V}{\partial x}\right) =q\left( p-n+N_{B} \right) $$ First set of normalization $$ \frac{qV}{kT}=u,\qquad\qquad n=n_{i}e^{u-...
3
votes
1answer
32 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 (...
1
vote
1answer
76 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 ...
4
votes
1answer
58 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} - \...
1
vote
0answers
27 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 ...
7
votes
1answer
134 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 ...
3
votes
1answer
87 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{\...
3
votes
1answer
111 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 ...
1
vote
3answers
118 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 ...
5
votes
1answer
167 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 ...
1
vote
0answers
42 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 ...
0
votes
0answers
17 views

Converting events with irregular arrival times into a continous signal (1st, 2nd and 3rd order)

I got events coming in at irregular intervals. From these I need to emit the number of events arrived and the expected first and second derivative over some specified measurement period P. So ...
0
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0answers
55 views

How to numerically determine the temporal accuracy of a discretization scheme?

By calculating the norm of the difference between a numerical and exact solution: $$\epsilon = \sqrt{\frac{1}{N}\sum_{i=1}^N\left(u_{i,num}-u_{i,exact}\right)^2}$$ and plotting this versus several ...
1
vote
1answer
171 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 ...
0
votes
1answer
80 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 ...
0
votes
0answers
53 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 ...
1
vote
1answer
104 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 ...
0
votes
0answers
56 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 ...
3
votes
2answers
89 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 ...
1
vote
0answers
100 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 ...
0
votes
0answers
22 views

Brownian motion noise strength in discrete time step and in continuous time

In this Langevin dynamics tutorial The second part is talking about Implementation. It says because we are using discrete time step, we need to divided the variance by time step. In Langevin ...
0
votes
1answer
40 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 ...
2
votes
0answers
80 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 ...
1
vote
1answer
64 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 ...
2
votes
2answers
114 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?
1
vote
2answers
197 views

Dirichlet BCs - alternative implementation methods

I am having problems with solving a hyperbolic wave problem with Dirichlet BCs. I have tried reducing the time step sizes, which does not affect the results, and notices increasing the number of nodes ...
2
votes
2answers
116 views

upwind schemes for solving inviscid euler equations

I'm new to the modelling of inviscid euler equations. I have come across few different upwind schemes that are used instead of central differencing schemes to model such flows, such as flux vector ...
1
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0answers
45 views

should boundary conditions be effecting moving mesh results?

I have a question on the use of moving mesh to solve the inviscid euler equations. I have solved the following equations: $$\frac{\partial}{\partial t}\left[\begin{array}{c} \rho\\ \rho u \end{array}\...
1
vote
1answer
121 views

FVM - virtual node discretisation

I have come across the paper titled: A monolithic fluid structure interaction algorithm ... For a 1D grid, at the boundaries the paper uses virtual nodes $x_{0}$ and $x_{N+1}$ (page 372) and for ...
1
vote
2answers
84 views

Building minimization optimization problem for 2nd-order elliptic PDE

I am solving elliptic PDE problem, for which, Euler scheme looks as following: $$ \nabla [\gamma ( |\nabla u|^2) \nabla u] = 0,$$ where $$\gamma(|\nabla u|^2) = (1 + |\nabla u|^2)^{-1/2}. $$ I am ...
1
vote
0answers
111 views

Minimal surface finite differences problem - Matlab assemble

I face to the following problem: $$(1+u_x^2)u_{yy} - 2u_xu_yu_{xy} + (1+u_y^2)u_{xx}=0.$$ Problem needs to be discretized and assembled. Does anybody know how to proceed in Matlab?
0
votes
0answers
98 views

How to compute this double integral?

Let $$T=1, K=100, S_0=100, \sigma=0.05, r=0.15. $$ Define $\nu:=\frac{2r}{\sigma^2}-1$ and $$H(y,z)=\frac{z e^{\pi^2 /4y}}{\pi \sqrt{\pi y}}\int_0^{\infty} e^{-z \cosh(u) -u^2/(4y)} \sinh(u) \sin(...
2
votes
0answers
308 views

Finite difference scheme for solving nonlinear least-squares problem

I am dealing with following problem: $$ \min_{u,\gamma}\Bigg\{ \frac{1}{1000} \iint_{S_2} {\gamma (x,y)^2 dxdy} + \iint_{S_2} {[u(x,y) - u_0 (x,y)]^2 dxdy} + \iint_{S_2} {[\Delta u(x,y) - \gamma (...
2
votes
1answer
184 views

Finite difference for nonlinear system of equation

\begin{equation} \frac{\partial C_i}{\partial t} = D_i \nabla^2 C_i - \frac{I \cdot \nabla t_i}{z_i F} - \sum_{i'} \frac{z_{i'}}{z_i} D_{i'}\nabla \cdot (t_i\nabla C_{i'}) \end{equation} \begin{...
2
votes
1answer
210 views

Computer Build for Scientific Computing

I am currently a .NET software developer(SQL Server, ASP.NET, C#, MVC & Web Forms). In my spare time I'm researching different areas of environmental science. E.g.(Hydrology, Ecology, Atmospheric ...
4
votes
1answer
57 views

Estimating the local compression/expansion ratio for a transformation on a point cloud

Let's say we have an unorganized point cloud P1 with N points, each with coordinates {x,y,z}. We apply non-rigid transformation to P1 (translation + rotation + warping), to obtain point cloud P2. ...
8
votes
2answers
237 views

Space-time finite element discretization for time-dependent PDEs

In FEM literature, semi-variational methods are typically used in the solution of time-dependent PDEs. I have not seen a fully-variational approach i.e. where space and time are discretised by FEM, ...
3
votes
1answer
74 views

communication penalty when using wide stencils in parallel computations

When reading about discontinuous Galerkin methods one finds the argument that these methods allow higher-order accuracy while maintaining a compact stencil (a cell only communicates with its direct ...
1
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0answers
186 views

Equilateral triangle based mesh generation by intersection

In work I am currently working on I need to mesh some structure with equilateral triangles to study it using a kind of discrete element method known as spring networks or Lattice model. To mesh the ...