Questions tagged [numerical-analysis-of-pde]
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96
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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 \...
-2
votes
0
answers
349
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Stokes equation and Helmholtz decomposition
I apologize in advance for the long question, but it involves some work I been thinking about and would like help with. The Navier-Stokes equation in $\mathbb{R}^3$ subjected to no gravitational ...
2
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0
answers
54
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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\...
- 131
3
votes
1
answer
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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 ...
- 33
1
vote
0
answers
282
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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 $(u,v)\in [0,a]^2$.
$$
\begin{cases}
\tau_u&=F\left( \gamma,\gamma_u,\gamma_v,\...
- 182
3
votes
2
answers
382
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Inconsistency in determinability of the solution of a linear first order PDE
Consider the following differential equation:
$$\frac{\partial u(x,t)}{\partial t} = - \frac{\partial u(x,t)}{\partial x} + u(x,t) \label{1}\tag{1}$$
with $u(x,0)=f(x)$. The solution of \eqref{1}, ...
- 350
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votes
2
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453
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What are dissipative PDEs?
I often come across the term dissipative (partial) differential equation in mathematical articles, especially in the context of hypocoercivity and entropy methods. I now have an intuitive idea of ...
- 63
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0
answers
34
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Studying the evolution of laplacian in NS equation
The Navier-Stokes equation in $\mathbb{R}^3$ subjected to no gravitational forces are provided by:
\begin{equation}\label{Eq1}
\dfrac{\partial }{\partial t} \textbf{u} + \left(\textbf{u}\cdot \nabla \...
- 185
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0
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69
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Where can I find the paper by Tappert and Hardin on split-step Fourier transform method?
The split-step method is a numerical method that can be used to solve a nonlinear PDE (https://en.wikipedia.org/wiki/Split-step_method). Even Wikipedia does not refer to the original authors (F.D. ...
- 101
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51
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Error estimates for inhomogeneous semidiscrete PDE
I have the following semidiscrete problem on a meshed domain $U_h$. Let
$V_h$ be linear finite elements on $U_h$, $V_{h0}\subset V_h$ have zero trace on $\partial \Omega_h$, and
$V_{h\partial}$ be ...
- 165
3
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0
answers
66
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Confusion with implementation of PDE constraint Bayesiain inverse problem
Consider a PDE,
$$\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(...
- 81
1
vote
1
answer
100
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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 ...
- 31
2
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0
answers
24
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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 ...
- 511
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0
answers
183
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Numerical Method Simulation for 2D Advection Diffusion Equation on Python [closed]
Here it is an Advection-Diffusion equation in 2D:
$$
\frac{\partial C}{\partial t}+U \frac{\partial C}{\partial x}+V \frac{\partial C}{\partial y}=D\left(\frac{\partial^2 C}{\partial x^2}+\frac{\...
0
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0
answers
37
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Rigorous definition of space and time order of accuracy of numerical PDEs
Suppose that we are solving numerically a PDE (with a numerical scheme like this one) which involves space $x$ and time $t.$ It is a commonly seen expression in the literature that "the method ...
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0
answers
23
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Generating a proper finite difference scheme
I have recently started studying the finite difference schemes for numerical analysis. While I can now calculate difference schemes fairly easily for simple equations, I've recently come across a ...
2
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0
answers
90
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Approximating solutions to Monge-Ampere from optimal transport plans
I am interested in finding numerical solutions to a Monge-Ampere type equation for applications in physics. Due to the close connection between Monge-Ampere and optimal transport and the well ...
- 705
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66
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Computing and isotopy of curves in $\mathbb{R}^3$
Imagine a piece of string in the ocean moving gently with the currents; the string bends but does not change its length.
The (stationary) string can be modelled by a unit speed curve:
$$[0,1] \...
- 171
1
vote
1
answer
156
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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
$$\dfrac{\partial}{\partial t}\...
- 185
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votes
2
answers
545
views
How to compute $\sin(\frac{d}{dx})f(x)$?
Assuming $f(x)=e^{-x^2}$ for $x$ in $[-10,10]$, I have tried the following:
Fourier transform $\mathcal{F}$: $\frac{d}{dx}$ can be diagonalized as $\mathcal{F}^{-1} i\omega \mathcal{F}$. Therefore, $\...
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66
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Kolmogorov $\epsilon$-entropy, $n$-width, and $\epsilon$-capacity and applications
What is the relationship between Kolmogorov $\epsilon$-entropy, Kolmogorov $n$-width, and Kolmogorov $\epsilon$-capacity of a set $M$ in a metric space $X$? (The $\epsilon$-capacity here is the ...
- 131
1
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73
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Highy non-linear PDE involving directional derivative
Let the convolution of two function $f$ and $g$ be defined over $\mathbb{R}^3\times [0,\infty)$ as followed
\begin{equation}\label{ConvoDef}
\left(f*g\right)\circ(\textbf{x},t) = \int_{0}^{t}{\int_{\...
- 185
2
votes
2
answers
222
views
$H^s$ norm of non-integer power of functions
Let $ \Omega = \mathbb{T}^d (1 \leq d \leq 3)$ be the $d$ dimensional torus and $ u \in H^2(\Omega) $ be a complex valued function. For some $ 0 < \alpha < 1 $, let $ g(u) = |u|^\alpha u $.
My ...
- 105
1
vote
0
answers
44
views
Flux that can be represented by low and high resolution schemes
In the wiki page of Flux limiter, it writes:
If these edge fluxes can be represented by low and high resolution schemes, then a flux limiter can switch between these schemes depending upon the ...
- 111
2
votes
0
answers
79
views
Derivation of the Cahn-Hilliard PDE from the point of view of finite difference methods
Consider the Cahn-Hilliard equation
$$\frac{\partial c}{\partial t} = \nabla^2(f(c)-\varepsilon^2 \nabla^2 c)$$
defined on your favorite domain. I'm looking for a literature reference that formally ...
- 53
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votes
1
answer
100
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FEM based solution to parabolic problem
Consider the problem
$$
\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
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0
answers
106
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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\...
- 121
3
votes
0
answers
60
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 ...
1
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0
answers
37
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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 ...
- 2,505
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56
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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 ...
- 1,569
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0
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120
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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 = ...
- 43
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1
answer
90
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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 ...
- 111
3
votes
0
answers
60
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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} \...
- 833
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votes
0
answers
67
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 ...
- 21
3
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2
answers
346
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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} \...
- 1,168
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votes
0
answers
68
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 ...
- 161
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0
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105
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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{...
- 255
6
votes
1
answer
408
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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 ...
- 827
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0
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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 $$\sup_{t\in[0,\:\tau]}\left\|v(t,x)-v(t,y)...
- 131
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votes
1
answer
271
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Conditions on the velocity ensuring that a flow moves points along the boundary of a manifold
Let
$\tau>0$;
$d\in\mathbb N$;
$v:[0,\tau]\times\mathbb R^d\to\mathbb R^d$ be Lipschitz continuous in the second argument uniformly with respect to the first with $v(\;\cdot\;,x)\in C^0([0,\tau],\...
- 131
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vote
0
answers
74
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Show that the support of the shape gradient $\nabla\mathcal F(\Omega)$ is contained in $\overline\Omega$
Let $E$ be a $\mathbb R$-Banach space, $\Theta\subseteq C^{0,\:1}(E,E)$ be a $\mathbb R$-Banach space and $(T^{(\theta)}_t)_{t\ge0}$ denote the $C^1$-diffeomorphism from $E$ onto $E$ with $$T^{(\theta)...
- 131
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0
answers
132
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Non linear second order PDE involving max operator (Dynamic Programming)
I'm trying to solve the following Dynamic Programming equation in continuous time ($dt \rightarrow 0$)
$$ v(x,t) = \max\Big\{|x|\,,\,v(x,t)+dt\Big(v_t(x,t)+\frac{1}{2(t+1)}v_{xx}(x,t)\Big) \Big\} - \...
- 109
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votes
1
answer
375
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Weird claims and conclusions in "Introduction to Shape Optimization"
I'm trying to understand the notions of Euler and Hadamard derivatives of shape functionals. All the lecture notes and papers on this topic that I've found seem to build up on the books Shapes and ...
- 131
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vote
0
answers
109
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Von Neumann analysis on a finite difference hyperbolic scheme
I am doing a Von Neumann analysis on a staggered finite difference scheme (for Maxwell's Equations).
The finite difference scheme is:
$$
\mathbf{u}_v|^{n+2}_{i,j} - \mathbf{u}_v|^{n}_{i,j} = - A \frac{...
- 11
1
vote
0
answers
100
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Differentiation under the integral sign for a $L^1$-valued function (shape derivative)
Let
$d\in\mathbb N$;
$U\subseteq\mathbb R^d$ be open and $$\mathcal A:=\{\Omega\subseteq U:\Omega\text{ is bounded and open and }\partial\Omega\text{ is of class }C^{0,\:1}\};$$
$E:=\bigcup_{\Omega\...
- 131
3
votes
1
answer
347
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, ...
- 131
3
votes
1
answer
175
views
More important or relevant progress in discretizing hard problems in physics in last decade
This is a reference request, and soft question as companion.
I'm curious to ask, from an informative point of view, what about the more important progress in the goal to discretize hard problems in ...
1
vote
0
answers
34
views
Choice of finite element spaces in plasticity
I am planning to run numerical simulations in metal elastoplasticity (von-Mises yield condition with and without isotropic hardening). However, I am completely new to this subject and I am unsure ...
- 111
2
votes
1
answer
184
views
Discrete curve-shortening flow – numerical implementation
I need to investigate the properties of open curves which evolve according to the standard curve-shortening flow (Wikipedia link), but with fixed extremes as boundaries (si it should converge to the ...
- 31
0
votes
0
answers
68
views
Fokker-Planck equation with zero diffusion coefficients on the boundaries
I am currently working on a Masters project in which I will be looking at numerical methods for a specific PDE. The PDE that I will be considering is a Fokker-Planck equation on a unit hyper-cube.
...
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