Questions tagged [numerical-analysis-of-pde]

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What the scientists in numerical solutions of PDE are concerned recently? [closed]

I'm a new one in the numerical solution of PDE, mainly interest in NS equations, Convection-Diffusion equations and I know a lot of people are solving equations by ML, but I really don't like this ...
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31 views

Convergence of numerical scheme for HJB equation

Convergence of numerical scheme for HJB equation has been widely studied, the key paper is the Barles's one. Essentially, the convergence is guarenteed if the scheme is: Consistent Stable Monotony ...
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132 views

Relative bounds for vorticity

Write the vorticity equation as \begin{equation}\label{Eq20} \begin{split} \dfrac{\partial}{\partial t} v_i & = \biggl[|\textbf{v}|~|\nabla u_i|\cos(\beta_i)- |\textbf{u}|~|\nabla v_i|\cos(\...
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How to solve with FEM a semilinear elliptic equation?

I searched in many books regarding FEM how to solve semilinear elliptic equation, but I did not find too many things. They mostly treat linear and simple problems. For example in P.Ciarlet-The finite ...
0 votes
1 answer
137 views

Where can I find the paper by Tappert and Hardin on split-step Fourier transform method?

The split-step method is a numerical method that can be used to solve a nonlinear PDE (https://en.wikipedia.org/wiki/Split-step_method). Even Wikipedia does not refer to the original authors (F.D. ...
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77 views

Can we study concavity of vorticity equation?

The vorticity equation is well known given as \begin{equation}\label{Eq1} \dfrac{\partial}{\partial t}\textbf{v} + (\textbf{u}\cdot \nabla )\textbf{v} - (\textbf{v}\cdot \nabla )\textbf{u} = \nu \...
3 votes
1 answer
230 views

How to generate a random function with conditions?

The background is as follows: I consider the following differential equation $$\phi_{xx}+u\phi=\lambda \phi,\ \ \lambda=-k^2$$ where $u=u(x),\ \phi=\phi(x,\lambda)$, $\lambda$ is the spectral ...
2 votes
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64 views

Are there spectral Galerkin methods for PDE of the form $\partial_tu=\nabla\cdot f(\nabla u)\nabla u$?

Question is in the title. The nonlinearity due to the term $f(\nabla u)$ makes it difficult to directly apply the spectral Galerkin method as it can be done for PDE of the form $\partial_tu=\nabla\...
1 vote
0 answers
292 views

Solutions of a Gauss–Codazzi-like system of nonlinear PDEs

Consider the following system of PDEs for the dependent variables $\tau=\tau(u,v)$ and $\gamma=\gamma(u,v)$, with $(u,v)\in [0,a]^2$. $$ \begin{cases} \tau_u&=F\left( \gamma,\gamma_u,\gamma_v,\...
3 votes
2 answers
409 views

Inconsistency in determinability of the solution of a linear first order PDE

Consider the following differential equation: $$\frac{\partial u(x,t)}{\partial t} = - \frac{\partial u(x,t)}{\partial x} + u(x,t) \label{1}\tag{1}$$ with $u(x,0)=f(x)$. The solution of \eqref{1}, ...
7 votes
2 answers
570 views

What are dissipative PDEs?

I often come across the term dissipative (partial) differential equation in mathematical articles, especially in the context of hypocoercivity and entropy methods. I now have an intuitive idea of ​​...
1 vote
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35 views

Studying the evolution of laplacian in NS equation

The Navier-Stokes equation in $\mathbb{R}^3$ subjected to no gravitational forces are provided by: \begin{equation}\label{Eq1} \dfrac{\partial }{\partial t} \textbf{u} + \left(\textbf{u}\cdot \nabla \...
1 vote
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51 views

Error estimates for inhomogeneous semidiscrete PDE

I have the following semidiscrete problem on a meshed domain $U_h$. Let $V_h$ be linear finite elements on $U_h$, $V_{h0}\subset V_h$ have zero trace on $\partial \Omega_h$, and $V_{h\partial}$ be ...
3 votes
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69 views

Confusion with implementation of PDE constraint Bayesiain inverse problem

Consider a PDE, $$\partial_t u -a \nabla u - ru (1-u) = 0$$ at a given snapshot in time. The inverse problem is to find the diffusion coefficient $a \in L^{\infty}$ from a noisy measurement $$Y = \Phi(...
1 vote
1 answer
105 views

How to numerically solve differential equations involving sines, cosines and inverses of the unknown function? [closed]

Crossposted at SciComp SE I'm very new to finite difference method and I am just introduced to methods of solving differential equation using finite difference method via sparse matrix method. I find ...
2 votes
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27 views

Free programs suggestions to simulate parabolic EDPs

I'm interested in learning how to computationally simulate the behavior of parabolic partial differential equations, but I don't know where to start, what are the best free programs to use and where ...
2 votes
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246 views

Numerical Method Simulation for 2D Advection Diffusion Equation on Python [closed]

Here it is an Advection-Diffusion equation in 2D: $$ \frac{\partial C}{\partial t}+U \frac{\partial C}{\partial x}+V \frac{\partial C}{\partial y}=D\left(\frac{\partial^2 C}{\partial x^2}+\frac{\...
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39 views

Rigorous definition of space and time order of accuracy of numerical PDEs

Suppose that we are solving numerically a PDE (with a numerical scheme like this one) which involves space $x$ and time $t.$ It is a commonly seen expression in the literature that "the method ...
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27 views

Generating a proper finite difference scheme

I have recently started studying the finite difference schemes for numerical analysis. While I can now calculate difference schemes fairly easily for simple equations, I've recently come across a ...
2 votes
0 answers
108 views

Approximating solutions to Monge-Ampere from optimal transport plans

I am interested in finding numerical solutions to a Monge-Ampere type equation for applications in physics. Due to the close connection between Monge-Ampere and optimal transport and the well ...
3 votes
1 answer
723 views

Navier-Stokes equations and machine learning

I am looking for a reference explaining how to solve the Navier-Stokes equations numerically using machine learning algorithms . Thank you in advance for your help .
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66 views

Computing and isotopy of curves in $\mathbb{R}^3$

Imagine a piece of string in the ocean moving gently with the currents; the string bends but does not change its length. The (stationary) string can be modelled by a unit speed curve: $$[0,1] \...
1 vote
1 answer
182 views

PDE involving curl

Let $G:\mathbb{R}^3\rightarrow\mathbb{R}^3$ be smooth vector field over $\mathbb{R}^3$. For which vector fields $F:\mathbb{R}^3\rightarrow\mathbb{R}^3$ does the PDE $$\dfrac{\partial}{\partial t}\...
4 votes
2 answers
552 views

How to compute $\sin(\frac{d}{dx})f(x)$?

Assuming $f(x)=e^{-x^2}$ for $x$ in $[-10,10]$, I have tried the following: Fourier transform $\mathcal{F}$: $\frac{d}{dx}$ can be diagonalized as $\mathcal{F}^{-1} i\omega \mathcal{F}$. Therefore, $\...
1 vote
0 answers
73 views

Highy non-linear PDE involving directional derivative

Let the convolution of two function $f$ and $g$ be defined over $\mathbb{R}^3\times [0,\infty)$ as followed \begin{equation}\label{ConvoDef} \left(f*g\right)\circ(\textbf{x},t) = \int_{0}^{t}{\int_{\...
3 votes
2 answers
355 views

Questions for the non-linear PDE $2u_t=\log(-u_{xx})$

Consider the PDE as follows : $$2u_t=\log(-u_{xx}), \quad \forall (t,x)\in [0,1)\times (-1,1)$$ with the terminal and boundary conditions $$u(1,x)=0,\quad \forall -1<x<1 \quad\quad \mbox{and} \...
2 votes
0 answers
82 views

Kolmogorov $\epsilon$-entropy, $n$-width, and $\epsilon$-capacity and applications

What is the relationship between Kolmogorov $\epsilon$-entropy, Kolmogorov $n$-width, and Kolmogorov $\epsilon$-capacity of a set $M$ in a metric space $X$? (The $\epsilon$-capacity here is the ...
2 votes
2 answers
269 views

$H^s$ norm of non-integer power of functions

Let $ \Omega = \mathbb{T}^d (1 \leq d \leq 3)$ be the $d$ dimensional torus and $ u \in H^2(\Omega) $ be a complex valued function. For some $ 0 < \alpha < 1 $, let $ g(u) = |u|^\alpha u $. My ...
1 vote
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52 views

Flux that can be represented by low and high resolution schemes

In the wiki page of Flux limiter, it writes: If these edge fluxes can be represented by low and high resolution schemes, then a flux limiter can switch between these schemes depending upon the ...
2 votes
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86 views

Derivation of the Cahn-Hilliard PDE from the point of view of finite difference methods

Consider the Cahn-Hilliard equation $$\frac{\partial c}{\partial t} = \nabla^2(f(c)-\varepsilon^2 \nabla^2 c)$$ defined on your favorite domain. I'm looking for a literature reference that formally ...
0 votes
1 answer
105 views

FEM based solution to parabolic problem

Consider the problem $$ \begin{cases} u_t - \Delta u = 0 &\text{ on } \Omega\times (0,T)\\u=0 &\text{ on } \partial \Omega\times (0,T) \\ u(x,0)=g(x) &\text{ on } \Omega \end{cases} $$ ...
2 votes
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115 views

Gradient $L^\infty$-estimate for heat equation with homogeneous Dirichlet boundary condition

$\Omega\subset \mathbb{R}^N$ is a bounded smooth domain. Consider the homogeneous heat equation with zero boundary condition in $\Omega$ \begin{cases} \partial_t u-\Delta u=0 \quad(x,t)\in \Omega\...
3 votes
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61 views

How I can distibute values over the computational cells?

I am an engineering student and I try to solve the fluid equations over a given set of computational cells. I have a mathematical question about a field I am currently studying, precisely the ...
2 votes
1 answer
264 views

Numerical methods for IDE [closed]

I would like to read a popular literature on the topic "Numerical methods for integro-differential equations". Could you recommend me any articles or book with a brief overview of some methods (maybe ...
1 vote
0 answers
40 views

P1-finite element as convolution of P0-finite element

For a vector $u\in\mathbf{R}^N$ let's denote $\pi_N(u)$ the unique piecwise linear and $1$-periodic function matching the components of $u$ on the discretization $x_k = \frac{k}{N}$ of the unit ...
1 vote
0 answers
56 views

How does a computer program recognize shocks given data of a solution to a conservation law?

Conservation laws are PDEs of the form $u_t +j_x=0.$ A discontinuous solution (for $u$ and $j$) to an equation like this can be easily found. Let's suppose that we are working with a piecewise ...
1 vote
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130 views

Semilinear PDE - BSDE presentation via Feynman Kac formula

For a semilinear PDE, we usually have this FBSDE representation: $\mathcal{X}_t=\mathcal{X}_0+\int^t_0 \mu (s,\mathcal{X}_s)\, ds\, +\int_0^t \sigma (s,\mathcal{X}_s)dW_s,\quad 0\leq t\leq T, \\ Y_t = ...
1 vote
1 answer
90 views

Typo in error a-priori estimate in a discontinuous Galerkin paper?

I'm looking at this famous paper which is available in the link below: Franco Brezzi, LD Marini, Endre Süli, Discontinuous Galerkin methods for first-order hyperbolic problems, Mathematical Models ...
3 votes
0 answers
60 views

Tuning parameters of PDEs given a set of data

I am interested in doing statistical inference in the context of PDEs. Loosely speaking, the kind of problem I have in mine is the following. Problem setting Let $(t_i, x_i, y_i) \in \mathbb{R} \...
2 votes
0 answers
69 views

Dense matrix vs sparse matrix, when they have same number of nonzero elements

I came across a new way in the literature to solve PDE problems numerically, which is called 'Patch Reconstruction'. One example paper is: Li, R., Sun, Z., Yang, F., & Yang, Z. (2019). A finite ...
5 votes
1 answer
106 views

Is the minmod limiter energy stable?

It is well-known, that upwind scheme and Lax-Wendroff scheme are energy stable for the linear advection equation $u_t +a u_x = 0$ with periodic boundary conditions, if the CFL condition is satisfied, ...
3 votes
0 answers
109 views

Smoothly connecting PDEs with finite differences

A PDE with non-smooth inhomogeneity Let $\mathcal{L}$ be a second-order, linear, elliptic differential operator acting on $\mathcal{C}^2([0,2]^2)$. I'm numerically solving the inhomogeneous PDE \begin{...
3 votes
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72 views

What is the purpose of converting a level-set function into a signed distance function?

In the paper Electrical impedance tomography using level set representation and total variational regularization, the authors tried to implement an iterative algorithm to find the interface of two ...
6 votes
1 answer
532 views

What is the big-O complexity of solving the sparse Laplace equation in the plane?

In MATLAB, you can get a 2d Laplacian via A = delsq(numgrid('S',N)); yielding a matrix $A$ that is $n \times n$ with $n = O(N^2)$, for a square domain discretized ...
0 votes
1 answer
264 views

Analytical testcase for 2D/3D anisotropic Diffusion (Heat Kernel)

I want to verify and compare different Discretizations of the anisotropic diffusion equation in 2D / 3D image of my testsetting. I want to verify and compare different Discretizations of the ...
1 vote
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86 views

Shape derivative at manifold $M$ in direction $v$ is equal to the shape derivative at $\partial M$ in drection $\langle v,n\rangle n$

Let $\tau>0$ and $d\in\mathbb N$. Definiton 1$\:\:\:$If $v:[0,\tau]\times\mathbb R^d\to\mathbb R^d$ with $v(\;\cdot\;,x)\in C^0([0,\tau],\mathbb R^d)$ and $$\sup_{t\in[0,\:\tau]}\left\|v(t,x)-v(t,y)...
3 votes
1 answer
273 views

Conditions on the velocity ensuring that a flow moves points along the boundary of a manifold

Let $\tau>0$; $d\in\mathbb N$; $v:[0,\tau]\times\mathbb R^d\to\mathbb R^d$ be Lipschitz continuous in the second argument uniformly with respect to the first with $v(\;\cdot\;,x)\in C^0([0,\tau],\...
0 votes
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134 views

Non linear second order PDE involving max operator (Dynamic Programming)

I'm trying to solve the following Dynamic Programming equation in continuous time ($dt \rightarrow 0$) $$ v(x,t) = \max\Big\{|x|\,,\,v(x,t)+dt\Big(v_t(x,t)+\frac{1}{2(t+1)}v_{xx}(x,t)\Big) \Big\} - \...
1 vote
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74 views

Show that the support of the shape gradient $\nabla\mathcal F(\Omega)$ is contained in $\overline\Omega$

Let $E$ be a $\mathbb R$-Banach space, $\Theta\subseteq C^{0,\:1}(E,E)$ be a $\mathbb R$-Banach space and $(T^{(\theta)}_t)_{t\ge0}$ denote the $C^1$-diffeomorphism from $E$ onto $E$ with $$T^{(\theta)...
3 votes
1 answer
376 views

Prove of the shape-derivative identity relating the shape and material derivative of a shape-dependent function

I've started reading about shape optimization. Most of the concepts I've encountered so far (such as the shape derivatives of domain and boundary integrals and the corresponding) seem to be complex, ...