# Questions tagged [discretization]

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

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### Time discretisation after splitting a 4th order equation

Suppose we have a fourth-order parabolic PDE $$\partial_t u + a(t)\Delta( \Delta u) - div( b(x,t)\nabla u) =0$$ We split the equation into two second-order equations by introducing $w = \Delta u$ thus ...
56 views

### What happens at the interface between a solid and a fluid?

I am doing an MPM simulation of water colliding against a solid, I am currently encoding the model for collision forces and I am realising something isn't clicking. Let us assume the fluid experiences ...
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1 vote
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### 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 ...
220 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 ...
118 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 $$\tag{1}u_{t} -6uu_x+u_{xxx}=0$$ I have ...
1 vote
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### Good non oscilliatory derivatives for an exsisting grid

I'm calculating the entropy production of a shockwave by utilizing the equations: \begin{equation} \sigma = J'_q\frac{\partial}{\partial x}\left(\frac{1}{T}\right) +\frac{1}{T}\frac{4\eta}{3}\left(\...
1 vote
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### 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 : https://github.com/mholtrop/...
1 vote
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### 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 ...
111 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: \begin{equation} \frac{\partial U}{\partial t} + \frac{\partial F(U)}{\partial x}...
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### 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 ...
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### 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
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### 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 ...
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### 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 ...
608 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 &= ...
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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### Why do we have to resort to Higher order schemes for solving the 1-D advection equation/ continuity equation?

\begin{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 ...
243 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. $$\frac{\partial u}{\partial t}+u(\nabla \cdot u)=-\frac{1}{\rho}\nabla p+\nabla ^2 u$$ ...
186 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
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1 vote
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### 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
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### 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 ...
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, $D(... 1 vote 1 answer 386 views ### Approximation Error in a Finite Difference Approximation of the Square of Derivative First Part: (First-order derivative) Assuming$f$is an infinitely differential function everywhere, the Taylor series of$f(x + h)$at$xis \begin{align}\tag{1} f(x + h) = f(x) + hf'(x) + \frac{1}... 0 votes 0 answers 83 views ### Analytic vs discrete understanding of PDE The PDE I am working with: $$\partial_tu = \nabla \cdot (a(x)\nabla u)-\beta(x)u\\ \partial_nu=0, x \in \Omega \subset \mathbb{R}^2\\ \beta(x)>0$$ Integrate the PDE: $$\int_\Omega \partial_t u=\... 0 votes 0 answers 38 views ### What will PDE discretization matrix look like for time and space? [duplicate] Please note: this question is not a duplicate of this question since, while the PDE is the same, the nature of this question is different, i.e. the other question treats a different aspect of this PDE.... 1 vote 0 answers 64 views ### Discretizing a parabolic PDE with finite volume method I want to discretize the following parabolic PDE:$$u_t = \nabla\cdot(\alpha(x)\nabla u)- \beta u\\ x\in\Omega \subset \mathbb{R}^2\\ \partial_n u = 0\\ u(t,0) = u_0(x)\ge 0, \alpha(x)>0$$Given ... 1 vote 0 answers 104 views ### Determine truncation error of PDE discretization The equation is$$\frac{\partial}{\partial x}\left(u\frac{\partial u}{\partial x}\right)=f(x)\\ 0<x<1, u(0)=u(1)=0$$I'm discretizing this PDE using FVM as follows: 0=x_0=x_{1/2}<x_1<x_{... 0 votes 0 answers 63 views ### Existence and uniquness of solution of FVM for Poisson equation I'm discretizing the following Poisson equation using FVM where the domain \Omega of the solution is a regular hexagon of side 1 centered about the origin.$$\Delta u =k,\text{k\$ constant}\\ \...
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: ...