# Questions tagged [discretization]

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

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### Looking for specific discretization operator

was reading notes related to NSe solving, trying to find the operator that discretizes diffusion term as follows: \begin{aligned} \nabla^{2} \mathbf{u} & \rightarrow\left[\begin{array}{cc} -\frac{...
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### Is there one general approach to build a projection methods for different problems?

My question is probably going to be too general to answer it with a couple words. Could you please suggest a good reading in that case. Projection methods are used to reduce size of the solution space ...
1 vote
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### finding discretization error in Burger equation

I was reading the paper given in the link http://www.unige.ch/~hairer/preprints/parareal.pdf and I have a problem in understanding in page 10 for the Burger equation on implementing Parareal method ...
1 vote
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### 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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### 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 ...
1 vote
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I come across the following operator in a paper $\mathcal{I}\psi = \psi_{xxxx} + (r~\psi_x)_x$, where $\psi=\psi(x)$ and $r=r(x)$. Periodic boundary condition is employed. It claims that the operator $... 1 vote 2 answers 79 views ### Can I use Q0 finite elements when there are gradients involved? Could I apply a Q0-discretization to, say, the Poisson equation$\Delta \phi = f(where by Q0 I mean piecewise constant, and thus non-continuous, elements)? Solving this with FEM, at least as I know ... 5 votes 1 answer 121 views ### Are Python/MATLAB/Mathematica numerical eigenvectors affected by eigenvalue degeneracies outside region of calculation? I have a discretized 2D mesh over which I calculate eigenvalues and eigenvectors of some Hermitian 2 x 2 matrix at each point along a closed loop parameterized by parameter t. The eigenvectors are ... 0 votes 0 answers 42 views ### P1 Finite element discretisation of laplace-neumann eigenproblem I am looking for help in the FE discretisation of the Laplace eigenkproblem with Neumann boundary conditions; that is, $$-\int_{\Omega} \nabla u \cdot \nabla v= \lambda \int_{\Omega} uv,$$ or $$Ax=\... 7 votes 1 answer 206 views ### Projection method FVM poisson part, adding source term The idea of the method is to decompose the Navier-Stokes equation into the solenoidal and irrotational parts.$$\frac{\partial u}{\partial t}+u(\nabla \cdot u)=-\frac{1}{\rho}\nabla p+\nabla ^2 u$$... 7 votes 1 answer 5k views ### What is numerical damping in the context of time-dependent FEM solvers? Comsol Multiphysics (a popular FEM package) includes two time-stepping algorithms (IDA aka BDF, and Generalized-alpha), described in their documentation as follows (quoted here under Fair Use; ... 4 votes 2 answers 125 views ### Three steps of pde numerical solution and nonlinear equation I'm trying to solve a nonlinear elliptic equation$$(n(u)u')' = f(u)$$and have a crucial misunderstanding. I suppose the procedure of solving some nonlinear equation consists of: Choosing a proper ... 0 votes 1 answer 73 views ### Time & Space matlab discretization Finite Differences confusion I have been trying to solve this equation and write the finite difference scheme in matlab for months, but I still am not successful. Given the KdV Equation$$\tag{1}u_{t} -6uu_x+u_{xxx}=0I have ... 0 votes 0 answers 39 views ### Gridline cutouts after FFT and iFFT on Python EDIT: I think I messed up on the coordinates of (p,q). Num was missing a multiple of 2\pi/N. Assuming my interpretation of DFT isn't wrong. I am currently using FFT to run Fresnel Diffraction as ... 1 vote 0 answers 254 views ### How to compute the Eigenvalue and Eigenstates of Quantum well with Effective mass using finite difference method in Python? I want to compute the eigenvalues and eigenstates of a quantum well with different effective masses of electron in the barrier and in the quantum well. As can be seen [1]: https://github.com/mholtrop/... 1 vote 0 answers 60 views ### Good non oscilliatory derivatives for an exsisting grid I'm calculating the entropy production of a shockwave by utilizing the equations: \sigma = J'_q\frac{\partial}{\partial x}\left(\frac{1}{T}\right) +\frac{1}{T}\frac{4\eta}{3}\left(\... 1 vote 1 answer 85 views ### Discrete model of cell - cell communication I am trying to understand how cell to cell communication is studied using a discrete modelling framework. Could someone please suggest suitable references or libraries which already have ... 3 votes 1 answer 98 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: \frac{\partial U}{\partial t} + \frac{\partial F(U)}{\partial x}... 2 votes 1 answer 73 views ### Dividing a continuous domain into small squares; how to perform storage and querying? I recently had a software engineering interview and was asked a series of questions that was a bit outside of knowledge realm, and I feel like there's some scientific computing principles here (I took ... 3 votes 1 answer 78 views ### Quantify difference between two discrete 1D solutions I have an ordinary differential equation that is solved as an initial value problem using different numerical schemes. I end up with several discrete time signals that should display a reasonably ... 1 vote 0 answers 41 views ### Procedure to convert continuous equations of motion to discrete version Let's take a mobile robot and suppose we know its continuous equations of motion, for example this car-like simple model. Now if I am simulating this robot in a continuous 2D plane, then coding the ... 0 votes 0 answers 55 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 ... 2 votes 1 answer 320 views ### Tensor product representation for the 9-point finite difference approximations for the Poisson equation If we use 5-point finite difference approximations in a uniform rectangular grid to solve the Poisson PDE \begin{align} -\Delta u &= f \ \ \text{en} \ \ (0,1)\times (0,1) \label{P1} \\ u &= ... 4 votes 1 answer 93 views ### Why do we have to resort to Higher order schemes for solving the 1-D advection equation/ continuity equation? \begin{aligned} \frac{\partial N}{\partial t} &+ \frac{\partial J}{\partial r} = 0, \\ \frac{\partial N}{\partial t} &+ \frac{\partial }{\partial r}(N \upsilon ... 2 votes 1 answer 302 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.$$Problem needs to be discretized and assembled. Does anybody know how to proceed in Matlab? 0 votes 1 answer 293 views ### Elliptic PDE finite volume method with Dirichlet boundary condition I want to discretize the following equation using a Finite Volume Method$$\nabla \cdot (a(x)\nabla u)=f(x)\\x\in \Omega \subset \mathbb{R}^2 \\u_{|\partial\Omega}=g$$I'm using Voronoi cells here: ... 0 votes 1 answer 148 views ### Discretization of a non-linear ODE using FDM isn't grid indepenent I am trying to solve the ODE : \frac{d^2T}{dx^2} = \omega_1 T+\omega_2 T^2 + using different numerical methods. I have tried the following discretizations so far and none of them seem to be grid ... 1 vote 0 answers 56 views ### discretization of advection diffusion with variable coefficients I am looking for help to find a somewhat stable FD numerical scheme for the advection diffusion equation posed on a curve (x(r),y(r)). The equation becomes$$u_t=\alpha(r) u_r +\beta(r) u_{rr}+f(r,t)... 1 vote 1 answer 73 views ### Rate of convergence - Stochastic Euler Method The absolute error criterion of the pathwise approximation of an Ito processX$by an Euler approximation$Yis: $$\epsilon=E\left(\left|X_{T}-Y(T)\right|\right)$$ We shall say that a time-... 1 vote 1 answer 92 views ### Discretization of a nonlinear boundary value problem I am trying to use finite element method to discretize the following problem \begin{align} \min_{u \in H^1_0(\Omega)} \int \| \Delta u(x) - 0.5*[u(x) + \langle e, x \rangle + 1]^3 \|^2_2 \ d\Omega, \... 2 votes 1 answer 148 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 ... 3 votes 1 answer 158 views ### FEM with elastic inhomogeneous properties leads to mesh-induced anisotropy I'm solving an elastic homogenization problem and I'm having problems with mesh artifacts. I would like to first give a brief summary of what I do: I have a system with inhomogeneous (but isotropic) ... 2 votes 0 answers 49 views ### In sights into why higher order finite differencing method leads faster to instability I was playing around with numerically solving the 1D wave equation with density and stiffness varying with position using central differencing methods and noticed that for certain discretization steps ... 3 votes 1 answer 208 views ### Choosing an appropriate time step for a discrete & continuous dynamics simulation I have inherited of a flight dynamics simulation in C++ which represents a small drone with it's autopilot, actuator dynamics and a solid state IMU. Hence, it is composed of a few models, some ... 0 votes 0 answers 45 views ### How to solve for underlying function from discrete data set containing integral of that function New to Computational Science, I hope I'm on the right exchange network for this question. I have a time series data set that contains the sum of a source data set representing an exponential decay ... 2 votes 3 answers 318 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 ... 4 votes 1 answer 178 views ### Differences between Discrete Fourier Transform and Continuous Fourier Transform? I am trying to visualize the time dependence of a free particle given an initial wave-function using Python and I just wanted to know if I could use the in built FFT implementation from NumPy to find ... 5 votes 0 answers 69 views ### Why does the naive barycentric hodgestar fail? The discrete exterior calculus is defined first using circumcentric dual cells, because the primal and dual edges are orthogonal and thus the dual cells are convex. This leads to a diagonal hodge star ... 0 votes 0 answers 69 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 ... 6 votes 2 answers 323 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 ... 28 votes 6 answers 20k views ### How can I numerically differentiate an unevenly sampled function? Standard finite difference formulas are usable to numerically compute a derivative under the expectation that you have function values$f(x_k)$at evenly spaced points, so that$h \equiv x_{k+1} - x_k$... 5 votes 2 answers 297 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 ... 0 votes 1 answer 516 views ### Solving differential equation in Python with discretized variable coefficients I am trying to solve a differential equation with discretized variable coefficients which are calculated from a time serie. In this case the Runge-Kutta step size is fixed by the frequency in the time ... 1 vote 0 answers 60 views ### Can the standard multigrid performance be used for time-dependent PDEs? Consider a time dependent pde(i.e u(x,t)).I know when only space-coarsening is used the standard multigrid performance can be applied but what if instead we use only time-coarsening?Can we apply the ... 1 vote 0 answers 136 views ### FDM discretization of equation on the boundary In order to simulate the following equation using FDM $$u_t(t,x)-u_{xx}(t,x)=0, \quad (t,x) \in (0,1)\times (0,1)$$ $$(u_t(t,x)-u_{x}(t,x))\rvert_{x=0}=0, \quad t \in (0,1)$$ $$(u_t(t,x)+u_{x}(t,x))\... 1 vote 1 answer 271 views ### How to do Weierstrass-transform in MATLAB? I have a diagonalization problem. I have the eigenstates correctly, and I want to do a Gaussian-smearing (Weierstrass-transform) on them. So I have the wave functions (\Psi), and the continuous ... 1 vote 1 answer 95 views ### Is "Gradient Computation" in Finite Volume Discretization Really 2nd order accurate? Based on this, pp 245, we go through these steps to discretize a gradient statement, namely \nabla\phi: 1- Gauss theorem reads,$$ \int_V\nabla \phi dV = \oint_{\partial V}\phi dS $$2- Integral ... 2 votes 1 answer 53 views ### Discretization with non-constant matrix containg entries form unknown vector Consider a system of PDEs$$ \begin{cases} u_t = \nabla \cdot (D(u)\nabla u) + \frac{c}{K_U+c}u-ku\\ c_t = d_c\Delta c -\frac{\nu_U c}{K_U + c}u \end{cases}$$with some boundary conditions. Here,$D(...
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First Part: (First-order derivative) Assuming $f$ is an infinitely differential function everywhere, the Taylor series of $f(x + h)$ at $x$ is \begin{align}\tag{1} f(x + h) = f(x) + hf'(x) + \frac{1}...