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

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3
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1answer
102 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
110 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
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1answer
101 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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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 ...
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0answers
15 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 ...
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42 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 ...
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1answer
124 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
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1answer
71 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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46 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
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1answer
94 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
50 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
72 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 ...
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0answers
81 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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0answers
16 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
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1answer
34 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
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0answers
73 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
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1answer
61 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
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2answers
108 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?
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2answers
190 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
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2answers
97 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 ...
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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 ...
1
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1answer
99 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 ...
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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 ...
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0answers
100 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?
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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) ...
2
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0answers
304 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
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1answer
163 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} ...
1
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1answer
124 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
53 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
187 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, ...
2
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1answer
70 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 ...
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0answers
135 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 ...
2
votes
1answer
215 views

Laplacian discretization for parametric curves

I know how to compute the discrete Laplacian of a graph and of a mesh (the Laplace-Beltrami operator). Is there an analogous definition for the computation of the Laplacian of a parametric curve ? ...
2
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0answers
47 views

2nd order differences for modelling a physically discrete phenomenon?

Consider the diffusion of some unknown $R$ on a physically discrete, 1-D ring: x---------x---------x---------x n = 0 1 2 3 We let ...
4
votes
2answers
106 views

Adaptive mesh refinement algorithms and the difference between AMR and moving mesh

I'm working on my thesis and a part of it has to do with adaptive mesh refinement. As a computer science major, I'm not too familiar with this field. The best way I can put my knowledge of AMR is: I ...
2
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0answers
396 views

What is numerical damping in the context of time-dependent FEM solvers?

Comsol Multiphysics (a popular FEM package) includes two time-stepping algorithms (IDA aka BDF, and Generalized-alpha), described in their documentation as follows (quoted here under Fair Use; ...
0
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1answer
37 views

Non linear system of equations with discretization on k-space

I want to numerically solve the following system of differential equations at the steady state: \begin{equation} \begin{aligned} \frac{\partial \rho_{11\mathbf{k}}}{\partial t} =& ...
4
votes
3answers
897 views

Discrete Poisson Equation with Pure Neumann Boundary Conditions

I'm trying to implement the Helmholtz-Hodge Decomposition in 2D, which states that a vector field is composed by a rotational free component, a divergence free component and a harmonic component. ...
3
votes
3answers
273 views

Surface integration over a portion of an ellipsoid

I would like to perform a surface integration over a portion $D$ of an ellipsoid. A plane arbitrarily intersects the ellipsoid forming two sections, of which one is $D$. I do not know how I can ...
2
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0answers
114 views

boundary conditions with non-constant coefficients in cell centred finite volume method

Suppose am solving the heat conduction equation in 1d with Dirichlet boundary conditions. The thermal conductivity $k$ is a non constant function. So $-(k(x)u'(x))' = f(x)$ The value of $k$ enters ...
3
votes
2answers
1k views

2nd order centered finite-difference approximation of $u_{xy}$

The problem is to find a 2nd order finite difference approximation of the partial derivative uxy, where u is a function of x and y. Page 5 of this pdf I found does a centered difference approximation ...
4
votes
1answer
270 views

Which finite-element framework allows the easy implementation of HDG methods?

I have tinkered around with the implementation of hybridizable Discontinuous Galerkin (HDG) methods in DUNE-PDELab and DUNE-FEM for my PhD but since then I have not had the time to work on these codes ...
5
votes
2answers
124 views

Discretization method for a reaction dominated elliptic PDE

I'm working with an elliptic reaction diffusion PDE of the form $$-k\nabla^2u+cu=0$$ I've noticed that when the reaction term dominates over the diffusion (i.e. $c>>k$), the true (exact) ...
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0answers
114 views

discretizing $\frac{d^2}{dx^2}$

I am trying to discretize the differential operator $\frac{d^2}{dx^2}$ acting on $S^1 = [0,1]$ using finitely many points around a circle at $0, \frac{1}{N}, \frac{2}{N}, \dots, \frac{N-1}{N}$. Here ...
7
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2answers
200 views

Spectral Methods in time

I was reading up on Spectral Methods for PDEs. In all the descriptions I read, while the position component is approximated via a Fourier series or other methods, the time component is still ...
6
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1answer
288 views

When is discrete Fourier transform a good approximation to the continuous one?

I am looking for help in understanding the use of discrete Fourier transform as an approximation to continuous Fourier transforms. As an exercise, I considered a Gaussian $$f(x) = ...
6
votes
2answers
477 views

Peculiar error when solving the Poisson equation on a non-uniform mesh (1D only) finite volume method

I have been trying to debug this error the last few days I wondered if anybody has advice on how to proceed. I am solving the Poisson equation for a step charge distribution (a common problem in ...
3
votes
3answers
194 views

Pseudo-inverse of a discretized operator with a null space?

Is there a way to understand what happens when a singular operator is discretized and inverted using the pseudoinverse (say using the SVD Moore-Penrose pseudoinverse)? For example, if we discretize ...
3
votes
1answer
241 views

Should the Jacobian of a system of PDEs be calculated from the main equations of the discretised equation?

I am solving a coupled system of non-linear PDEs in 1D. Something like, $$ u_t = F_1(u,v,w) \\ v_t = F_2(u,v,w) \\ w_t = F_3(u,v,w) $$ where each variable is a function of $x$ (the spatial ...
3
votes
2answers
5k views

Time stepping in comsol multiphysics

I would like to know which is the algorithm Comsol uses in order to correct the time step it uses. For example, when you try to solve an equation you've written in the PDE coefficient interface, and ...