1
vote
0answers
110 views

Runge-Kutta with all nodes at n+1 or zero weights otherwise [migrated]

So, lets say for the family of the explicit Runge-Kutta methods: $$y_{n+1} = y_n + \sum_{i=1}^s b_i k_i$$ where, $$k_1 = hf(t_n, y_n)$$ $$k_2 = hf(t_n+c_2h, y_n+a_{21}k_1)$$ $$\vdots$$ $$k_s = ...
2
votes
0answers
104 views

numerical method (implicit , backward difference or forward difference) for nonlinear pde

$\newcommand{\lbar}{\underline{\lambda}}$ In this linear PDE: \begin{cases} B_t+b^Q(r,t)B_r+\frac{1}{2}d^2(r,t)B_{rr}+(\mu(\lambda,t)+\alpha \sigma (t))(\lambda -\lbar)B_{\lambda} \\ ...
0
votes
0answers
41 views

Why Does a quadratic phase term in BNLS causes collapse?

I've heard a couple of times that in the Biharmonic Nonlinear Schrodinger Equation, $i\psi_z + \Delta ^2 \psi + |\psi | ^{2\sigma } \psi =0 $, $\psi (x, 0) = \psi _0 (x) \in H^2( \mathbb{R} ^d ) $ ...
1
vote
3answers
164 views

What are some good sanity checks for simulating BNLS?

After doing some googling, I failed to find any explicit solution for the Biharmonic Nonlinear Schrodinger Equation, which states: $$ i\psi (x,t) _t - \Delta ^2 \psi (x,t) + |\psi (x,t) | ^{2 \sigma} ...
0
votes
0answers
36 views

How can one use stability analaysis of finite differences methods in linear Schrodinger to the NLS?

Specifically, I've seen a lot of analysis of grid stability for solving Linear Schrodinger with Forward Euler, Backward Euler and Crank-Nicolson. However, most of the usages I've seen for the same ...
4
votes
0answers
265 views

Linearizing and solving a nonlinear PDE numerically

Im trying to solve the following (transport & diffusion) nonlinear PDE numerically (via finite volume on a cuboid region. Some Material gets cooled down, s.t. in some areas the material becomes ...
1
vote
3answers
257 views

What are the basis functions for a product space?

Let $X=L^1\left([0,1]^3\right)$, for numerical purpose, what are the possible basis function for $X$? In finite element method, the basis functions are tooth functions, or polynomial functions. Is ...
10
votes
1answer
319 views

Who is Petrov of the Petrov-Galerkin method?

I was not able to find the origin of the name Petrov in the Petrov-Galerkin method for the numerical approximation of PDEs. Wikipedia refers to a certain Alexander G. Petrov, but it is still not ...
1
vote
1answer
174 views

What are the advantage of using operational calculus for numerically solving pde compared to FE or FD?

For numerically solving a partial differential equation (PDE) what advantage does operational calculus (OC) has over common methods like finite difference (FD), and finite element (FE)? I mean OC in ...
32
votes
2answers
1k views

Recent fundamental new directions in PDEs

My main interests are in modern geometry/topology, algebra and mathematical physics. I observe that there is a raising communication, language and social barrier between this community and the ...
22
votes
3answers
959 views

“Wild” solutions of the heat equation: how to graph them?

It has long been known that the Cauchy initial-value problem for the classical heat equation on $\mathbb{R}$ (or $\mathbb{R}^n$) doesn't have unique solutions, without additional assumptions. In ...
1
vote
0answers
71 views

Orthogonal projection of discontinuous piecewise polynomial space in energy scalar product

Let $I = [0,1]$ be the unit interval Let $I$ be partioned into $n$ closed subintervals $(I_j)_J$, each of length $1/n$. Let $X_{DC} = \{ v \in L^2[0,1] | 1 \leq j \leq n : v_{|I_j} \in \mathcal P_1( ...
2
votes
0answers
198 views

Convergence rate of iterated nonlinear equations?

For $i=1, \dots, n$ ($n$ could be large) we have variables $x_i$ and $y_i$ relating to probability bounds s.t. $x_i, y_i \geq 0, x_i+y_i \leq 1 \; \forall i$. Each $i$ has a constant $\theta_i$, and ...
4
votes
1answer
498 views

Solution of Helmholtz-Equation where Phase is restricted by additional PDE

Hello! Let's say I have $(\partial_x^2 + \partial_y^2 + a)f(x,y)=0$ with $f(x,y) \in \mathbb{C}$, ($\lim_{x,y \to \infty} f(x,y)=0$). Now separate the Amplitude and Phase of the solution: ...
0
votes
0answers
55 views

Ways to decompose a torus for finite element method so that each cell contains a complete revolution of the major radius

I've got a finite element problem involving paths around the interior of a torus. For this particular problem I think I could make things more computationally efficient if each cell in the mesh made ...
0
votes
1answer
234 views

Discrete Sobolev space of $R^n$ valued maps

Can some one tell me the reference or any idea how to take the Discrete Sobolev space work defined for a scalar valued map to the space of maps which are vector valued.Let's say $f:\Omega ...
2
votes
0answers
285 views

How to apply Lagrange Multipliers to BCs of Time Dependent problems using finite elements?

I am trying to implement a finite element scheme using the method of lines (finite difference in time and finite element in space) and enforcing boundary conditions using Lagrange Multipliers. This ...
1
vote
1answer
138 views

What we mean by positive solution and radial solution of any partial differential equation

I am not getting the most of the article concern with the existence and uniqueness of positive solution and radial solution. I just want to know how a positive solution and radial solution of ...
2
votes
0answers
270 views

Variational Formulation of Boundary Value Problems With Unknown on the boundary

Suppose that we have a linear operator equation on $\Omega$ with Lipschitz boundary $\partial \Omega$, \begin{eqnarray} Lu &=& \frac{\partial u}{\partial t}, u(x,0) &=& u_0 \; \; ...
4
votes
1answer
477 views

Use of games to approximate solutions to Partial Differential Equations

Hi there, Hopefully the mathematics community can help me out this one, I'm currently studying my senior capstone at my college, and decided to do some research on a chapter in Stanley Farlow's book ...
1
vote
0answers
311 views

Definition of spectral gradient

Consider this differential operator $$ \mathcal{H}(\phi(\mathbf{x})) = -\triangle + V(\mathbf{x})H_\epsilon (\phi(\mathbf{x})) $$ where $\mathbf{x} \in \mathbb{R}^2$, $\phi : \mathbb{R}^2 \rightarrow ...
19
votes
3answers
1k views

Convergence of finite element method: counterexamples

There are many known results proving convergence of finite element method for elliptic problems under certain assumptions on underlying mesh [e.g., Braess,2007]. Which of these common assumptions are ...
2
votes
0answers
455 views

Comparision of cubic hermite finite element and cubic B-spline finite element (in condition nunmbers of stiffness matrix, or sth else)

Background Consider the one dimensional second order elliptic PDE, $$ \left\{\!\! \begin{aligned} & -(a(x)u'(x))'+b(x)u(x)=f(x)\qquad x\in[0,1]\\ & u(0)=u(1)=0 \end{aligned} ...
3
votes
0answers
176 views

Numerical solution

Last time, I asked this question but after discussing with some friends, I have given up finding closed-form solutions. Now I have a simpler question.Let $g_i: i=1,2$ be $C^2 =C^{2}(-\infty,\infty)$ ...
2
votes
0answers
264 views

A free boundary problem by finite difference method

I wanna discretize the following free boundary problem Find $u$ and $\Omega$ such that $\Delta u=1-\delta_0$ in $\Omega$ with the conditions $u=|\nabla u|=0$ on $\partial \Omega$. I apply finite ...
1
vote
1answer
595 views

Tangential boundary conditions for magnetostatic FEM problem

Hi everybody, I am trying to solve a magnetostic problem with the Finite Element Method. But I have a problem applying tangential boundary conditions for the magentic field. I solve for the vector ...
0
votes
1answer
487 views

Delta notation used for describing numerical stencil

While reading some papers translated from the Russian literature, I've noticed that a delta symbol can be used to denote a FDTD stencil that discretizes a PDE. For example, in [1], a fourth order ...
1
vote
0answers
384 views

Splitting wave equation for application of CPML

A recent paper (Roden and Gedney, 2000) proposed the application of a Convolutional Perfectly Matched Layer (CPML) to approximate free-field conditions for Finite-Difference Time-Domain (FDTD) ...
2
votes
2answers
676 views

Application of coordinate-stretching transformation for Perfectly Matched Layer

A Perfectly Matched Layer (PML) is an absorbing boundary condition (ABC) which can be used to approximate free-field conditions for the numerical solution of wave equation problems. PML note The PML ...
1
vote
1answer
1k views

Boundary conditions of wave equation near infinity

For the following wave equation $ \frac{{\partial ^2 p}}{{\partial ^2 x}} + \frac{{\partial ^2 p}}{{\partial ^2 y}} = A\frac{{\partial ^2 p}}{{\partial ^2 t}} + B\frac{{\partial p}}{{\partial t}} $ ...
3
votes
1answer
1k views

Is there a general method for determing the domain of dependence of (higher-order) PDEs?

Is there a general method for determining the domain of dependence of (higher-order) PDEs? I would be plenty happy with a reference to a paper or textbook I could look at; despite much reading ...
4
votes
3answers
1k views

Numerical instability using only Heun's method on a simple PDE.

I'm trying to simulate the evolution of the Wigner function (a pseudo probability distribution over phase space) for a point particle moving in a chaotic potential. I'll provide background first, as ...