Questions tagged [numerical-analysis-of-pde]
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54
questions
3
votes
1answer
105 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
0answers
26 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 ...
2
votes
1answer
58 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 ...
0
votes
0answers
39 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.
...
0
votes
0answers
35 views
Weak formulation of PDE with weighted inner product
In Boyd's book on spectral methods (available here: https://depts.washington.edu/ph506/Boyd.pdf), I stumbled in section 3.5 "Weak & Strong Forms of Differential Equations: the Use-fulness of ...
1
vote
0answers
117 views
Rigorous error estimate for semi-discrete heat equation in bounded domain
Let $\Omega$ be a bounded Lipschitz domain in $\mathbb R^N$ and $u_h$ be a solution of
$$
\begin{cases}
\partial_t u_h -\Delta_h u_h = f(x) & \text{ in } \Omega_h\\
u_h=0 &\text{ in } \...
0
votes
0answers
24 views
Reference request: numerical methods for HJB free boundary problems
Suppose $r: \mathbb{R}^{d+1}\to \mathbb{R}, \ g: \mathbb{R}^d \to \mathbb R,\ b: \mathbb{R}^{d+1}\to \mathbb{R}^d$ and $ \sigma: \mathbb{R}^d \to \mathbb R^d, d \ge 2$, and consider an optimal ...
2
votes
2answers
148 views
Representing a nonlinear elliptic PDE as an energy minimization problem
I need to solve a PDE in 2D representing a (time-independent) nonlinear diffusion process. The unknown function is $\phi(x,y)$ and its gradients create fluxes $\vec J$ through a nonlinear relation:
$$\...
0
votes
0answers
76 views
solutions of systems of first order linear pde with non-constant coefficients
I would like to find a reference that discusses, in certain generality, the properties of solutions of systems of first order linear pde with non-constant coefficients. I am actually studying the ...
3
votes
0answers
60 views
Shooting method still relevant?
I'm preparing to teach a "Numerical Analysis II" course next term, and in previous years this course involved a section on the shooting method for solving one-dimensional boundary value problems. This ...
3
votes
0answers
42 views
Spectrum of a symmetric saddle point matrix
Let $C=\left[ {\begin{array}{cc}
A & B^{T} \\
B & O \\
\end{array} } \right]$, where $A\in \mathbb{R}^{n\times n}$ is SPD, $B\in \mathbb{R}^{m\times n}$ and $m\leq n$. The matrix $B$ ...
1
vote
0answers
47 views
discrete Fourier transform for composition of differential operators on a grid
This question pertains to stability analysis of finite difference methods using the discrete Fourier transform.
Suppose I have a convection diffusion equation of the form:
(1) $\hspace{.5in}u_t + \...
1
vote
0answers
24 views
Discrete maximum priniciple for parabolic operators
While reading a paper on the topic 'Numerical solutions for generalized Black-Scholes equation', It is given that their numerical scheme can be executed explicitly by solving a linear system $\mathbf ...
13
votes
4answers
1k views
Is there a connection between representation theory and PDEs?
As a PhD student, if I want to do something algebraic / linear-algebraic such as representation theory as well as do PDEs, in both the theoretical and numerical aspects of PDEs, would this combination ...
3
votes
1answer
48 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, ...
4
votes
1answer
110 views
Numerics for continuity equation with Sobolev vector field
Has any work been done about numerical methods for the continuity equation
$$
\partial_t \rho(x,t) + \operatorname{div} (b(x,t) \rho(x,t)) = 0, \qquad t \in [0,T], \quad x \in \mathbb R^N,
$$
where $...
2
votes
0answers
76 views
How we can do the derivative for this equation w.r.t.to time t>0
Let $x\in[0,L]$ and consider the following equation,
$$\varepsilon \left( t \right)=\frac{1}{2}\int_{0}^{L}{({{\rho }_{1}}{{\left| {{\varphi }_{t}} \right|}^{2}}+{{\rho }_{2}}{{\left| {{\varphi }_{t}} ...
2
votes
1answer
165 views
Numerical methods for IDE [closed]
I would like to read a popular literature on the topic "Numerical methods for integro-differential equations".
Could you recommend me any articles or book with a brief overview of some methods (maybe ...
0
votes
0answers
82 views
Why does this numerical scheme work on this nonlinear PDE?
i am currently solving a nonlinear PDE of mixed parabolic/hyperbolic type of the Form
\begin{align*}
\frac{\partial}{\partial x} \left(A \frac{\partial p}{\partial x} \right)
+\frac{\partial}{\...
1
vote
1answer
186 views
Time discretization in the Feynman-Kac formula with boundary conditions
I am applying the Feynman-Kac theory for solving a PDE with boundary conditions.
For the SDE simulation I use the Euler-approximation, which introduces a time-step $h$ for the Brownian Motion, and ...
2
votes
1answer
128 views
Construct examples satisfying some inequalities [closed]
How do I construct two vectors $a,b\in \mathbb{R}^{n}$, $a=(a_1,a_2,\ldots, a_n)^T$ and $b=(b_1,b_2,\ldots, b_n)^T$ which satisfy in the following conditions
\begin{align}
& a_ib_i\geq 1,a_ib_j&...
1
vote
0answers
96 views
Error estimates in the $L^2$ norm of conforming FEM about Poisson’s equation with mixed boundary conditions
Consider Poisson’s equation
$$- \Delta u = f{\rm\qquad{ in }}\;\Omega $$
with following mixed boundary cconditions
$$u = g{\rm\qquad{ on }}\;\Gamma \subset \partial \Omega $$
$$\frac{{\...
2
votes
0answers
115 views
How to solve this integral equation?
$$g(p) = \frac{1}{p^3} + \frac{2}{p^2} \int_{p}^{1}{(u-p)g(u)du}$$
Need to find $g(p)$ for $p > 0$.
If there is no explicit solution, how to solve it numerically? Maple13 calculates bad results, ...
0
votes
1answer
94 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 ...
11
votes
3answers
558 views
Which matrices can be realized as the Dirichlet-to-Neumann map for a given domain?
Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $n$ electrodes (Dirichlet BC $u=\text{...
2
votes
1answer
353 views
Naviers Stokes equation and machine learning
I am looking for a reference explaining how to solve Navier-Stokes numerically using Machine learning algorithms .
Thank you in advance for your help .
3
votes
1answer
88 views
Numerical iterative methods for Poisson equation
Given a domain $\Omega \subset \Bbb R^n$ and $\Delta\varphi=f$ where $\varphi:\Bbb R^n \to \Bbb R$ is unknown and $f:\Omega\to \Bbb R$ is a blackbox function (for each $\bf x$ it provides $f({\bf x})$,...
2
votes
0answers
51 views
$L^{1} $ estimate for $2^{nd}$ order elliptic boundary value problem
This is probably a classical question in numerical analysis of PDE (but I don't know the answer).
Suppose you are solving a traditional elliptic problem, for example, $u\in H^1_0(\Omega)$ (in a nice ...
2
votes
2answers
93 views
Iterative method for $p$-Laplacian
Consider the following iterative procedure for solving the $p$-Laplace equation $\nabla \cdot (|\nabla u|^{p-2} \nabla u) = 0$ with fixed Dirichlet boundary data:
$u_0$ is our initial guess, for ...
2
votes
1answer
97 views
Simulating Fisher Equation (FKPP)
I'm researching about microbial growth (on 2D). I think that a microbial population can be modeled by the Fisher equation (any other suggestion is welcomed). My doubt is about how can I solve ...
1
vote
0answers
123 views
A discrete-to-continuous approach to the Dirichlet principle?
Dirichlet principle: Let $\Omega \subset R^n$ be a compact set with $C^1$ boundary. Then, there exists a unique solution $f$ satisfying $\Delta f = 0$ in $\Omega$ and $f=g$ on $\partial \Omega$.
We ...
6
votes
0answers
275 views
Steklov eigenvalue problem for a planar region bounded by ellipse
The Steklov problem for a compact planar region $\Omega$ is
\begin{cases} \Delta u =0 &\text{in $\Omega$}, \\ \frac{\partial u}{\partial n} = \sigma u &\text{on $\partial \Omega$},
\end{...
2
votes
0answers
56 views
Coarse grid correction
Let $A_h \in \mathbb{R}^{n \times n}$ be the matrix corresponding to a finite element discretization of some nonselfadjoint, bounded and indefinite bilinear form corresponding to a second order ...
1
vote
1answer
140 views
A mathematical motivation for Lax-Friedrich type of Numerical Fluxes
A Lax-Friedrichs (LF) type of flux for a conservation law $\partial_tU+\partial_xf(U)=0$ is given by
\begin{align}
F(U^-, U^+) = \frac{1}{2} \Big(f(U^-) + f(U^+)\Big)\cdot \nu - \frac{1}{2} \lambda(U^...
2
votes
2answers
107 views
Solving numerically an equation involving exponentials [closed]
I met an equation of the following form:
$$\sum_{i=1}^nk_ip_i e^{-k_i\lambda}~~=~~b,$$
where $p_i\ge 0$, $k_i$ and $b$ are known for $i=1,\cdots, n$. I'd like to know how to find the solution $\...
0
votes
0answers
121 views
Time discretization of the variational formulation of the Navier-Stokes equation
Let
$T>0$
$I:=(0,T]$
$d\in\mathbb N$
$\Lambda\subseteq\mathbb R^d$ be nonempty and open, $$\mathcal V:=\left\{\phi\in C_c^\infty(\Lambda,\mathbb R^d):\nabla\cdot\phi=0\right\}$$ and $$V:=\overline{...
1
vote
0answers
57 views
Time discretization of the (stochastic) Navier-Stokes equation
Let
$d\in\mathbb N$
$\Lambda\subseteq\mathbb R^d$ be nonnempty and open
$\langle\;\cdot\;,\;\cdot\;\rangle:=\langle\;\cdot\;,\;\cdot\;\rangle_{L^2(\Lambda,\:\mathbb R^d)}$
I've found a thesis where ...
1
vote
0answers
64 views
Ask for reference about finite difference method on HJB equation
I am a fresh PhD student in numerical analysis. Recently I am considering finite difference methods and their error analysis for solving HJB equation of the following form:
$$
v_t=g(a(x)v_x),\quad x\...
3
votes
1answer
146 views
Methods to compute the Green's function for the 1D wave equation with nonsmooth coefficient?
I am seeking advice on the best available numerical methods to compute the Green's function for a 1D wave equation with rough coefficient.
Suppose that the coefficient $c(x)$ in the 1D wave equation ...
8
votes
1answer
2k views
Review paper/book on Finite Difference Methods for PDEs
I am looking for a good, relatively modern, review paper/book on Finite Difference Methods for PDEs with a theoretical emphasis in mind. By theoretical emphasis I mean that I care about theorems (i.e. ...
3
votes
1answer
220 views
Solving a differential system
Let $\mu$ be a probability measure on $\mathbb R$ with Lebesgue density, i.e. $\mu(dx)=\mu(x)dx$. We aime to find increasing and decreasing functions $\phi_{+}: \mathbb R_+\to \mathbb R_{+}$ and $\...
2
votes
1answer
107 views
Finite Element Method on a single triangular element
Consider the Laplace equation on a single triangular domain with a Dirichlet condition on two of the sides and a Neumann condition on the remaining side. I am using a linear element ... $\mathbb{P}_1$ ...
0
votes
1answer
243 views
A condition for Laplacian
Let $u\in L^{2}(\mathbb{R}^{2}) $ with $-\Delta(u) -c (x^{2}+y^{2})u \in L^{2}(\mathbb{R}^{2})$ where $c>0$.
Is true $-\Delta u \in L^{2}(\mathbb{R}^{2})$?
Thank you in advance.
0
votes
0answers
124 views
Fundamental solution matrix of a linear PDE
I've asked a very similar question also at math.stackexchange, but I've not received any answer.
A vectorial function $\boldsymbol{x}:\mathbb{R}^D \rightarrow \mathbb{R}^N$ satisfies the following ...
1
vote
0answers
145 views
A mathematical biology reference request
Is there any mathematical articles that describe the differential equation modelling of locomotion of amoeba using pseduopodia? I am looking for physics based models of pressure difference modeling of ...
1
vote
0answers
67 views
About the “method of lines”: when are such solutions good approximations for **all** future time?
This question is about approximate solutions to some classes of PDEs obtained using the "method of lines".
For example, for an initial-value problem given by a PDE on a circle, one can choose $n$ ...
2
votes
0answers
78 views
Properties of a Sobolev bound
I am interested in computing
$$
A:=\inf_{f\in L^{2}(\mathbb{R}^3)}\frac{||K^{\frac{1}{4}}f||_2^2}{||f||_{\frac{5}{2}}^2}
$$
where $K:=-\Delta+1$. We call $f_c$ the function that saturates the bound.
...
4
votes
0answers
575 views
What does the Von Neumann's stability analysis tell us about non-linear finite difference equations?
I've asked this question on computation science stackexchange, but it did not receive any answers so I have decided to ask it here as well.
I am reading a paper [1] where they solve the following non-...
2
votes
1answer
939 views
Solving a simple Schrödinger equation with Fast Fourier Transforms
While trying to solve a stochastic Gross-Pitaevskii equation I have found a problem that can be tracked down to something buggy occurring in the simplest Schrodinger equation possible:
$$\partial_t \...
3
votes
0answers
159 views
Numerical inversion involved confluent hypergeometric (1F1) (or Kummer function)
Edit: The question is solved !! The code is actually correct. There is not error in the codes. I miss-used it. Thank you for your attention : )
This problem arises when I tried to compute the valua ...
