0
votes
1answer
16 views

discretization of mixed boundary conditions in the advection diffusion reaction equation, the Crank Nicolson method

Sorry, I have some doubts regarding the discretization of mixed boundary conditions in a PDE. I have discretized my equation, but I doubt about the boundary conditions. I dont know if you have to ...
2
votes
1answer
28 views

Online resources to learn numerical methods for PDEs?

I would like to get into a career that uses alot of applied math. I took a numerical analysis course in undergrad and liked it, so I plan to self-learn numerical methods for PDEs. Other than the MIT ...
5
votes
0answers
40 views

Careers in applied math with an MS other than in finance and data/machine learning?

Since I like math, I would like a career that uses alot of applied math. I'm about to complete my Master's and could do my thesis in numerical solutions of PDEs I'm already aware of careers such as ...
2
votes
0answers
29 views

Finite difference scheme for hyperbolic system

I'm having a bit of trouble understanding the following, so it'd be great if anyone has any nice explanations! Thanks in advance! Consider the hyperbolic system $$u_t = Au_x + Bu$$ where $A$ and $B$ ...
2
votes
0answers
53 views

Compute average and maximum value of a field over a streamline

I'm working on a code solving a set of PDEs. I have a vector field, $\vec{v}(x,\theta,z,t)$ (it's a velocity) and a scalar field, $c(x,\theta,z,t)$. I have a $2\pi$-periodicity in $\theta$. The ...
0
votes
0answers
20 views

Accuracy of a finite-difference method for numerically solve a PDE or BVP

When solving the Poisson Equation $$-u''(x)=f(x)$$ with Dirichlet-Neuman boundary conditions $$u(0)=0, u'(1)=0$$ using a finite difference 2-order centered scheme and a 2-order upwind ...
1
vote
1answer
27 views

Problem on energy of a Discrete Galerkin Method

I'm reading an article from this website: article question is in page 3,about a wave equation,and use the Galerkin method to discrete the space. (1) page4 why the author use the fraction ...
0
votes
0answers
15 views

Finite difference method for elliptic pde stability condition

I am trying to implement a finite difference scheme for the elliptic pde $\nabla(a(x_1,x_2)\nabla\phi(x_1,x_2))=0$ where $(x_1,x_2) \in (0,l_1)\times(0,l_2)$ with $l_1 \neq l_2$ necessarily. I know ...
1
vote
1answer
34 views

Finite Difference Method Stability with diffusion equation

The diffusion equation is: $ \frac{\partial T}{\partial t} = \alpha \left( \frac{\partial^2 T}{\partial x^2} \right) $ An explicit finite difference approach can be used to solve this, forward in ...
0
votes
1answer
9 views

Change of variables from intinite to bounded support.

I may be missing something simple, but I am stuck. My question: I am solving a system of partial differential equations numerically, but one of the variables can take on any value, ie $x \in ...
1
vote
1answer
64 views

Function Approximation

I need to solve the following equation $$-\frac{\partial S(x,y,t)}{\partial t}=ax^2+bx\frac{\partial S(x,y,t)}{\partial x}+c\Big[\frac{\partial S(x,y,t)}{\partial ...
0
votes
0answers
36 views

On an interesting boundary condition

So I am tackling an interesting boundary condition, where $B(Du)=0$, for $x\in\Omega$, where $B$ is the signed distance function to $\Omega^*$ (where $\Omega,\Omega^*$ are convex domains in $\Bbb ...
2
votes
2answers
83 views

What do mathematicians mean by “analytical solution of an equation”?

Given a PDE equations of the form: $\dfrac{\partial}{\partial t} u(t,x) = \left(\hat{L}+\hat{N_u}\right)u(t,x) \;\;\;\;\;\;\hspace{10mm}(**)$ where $\hat{L}$ is a linear operator and ...
3
votes
1answer
47 views

Korteweg–de Vries equation: why is there a substantial literature on their numerical solutions if they are analitically integrable?

Given the initial value problems for the Korteweg-de Vries equation $u_t + u_{xxx} = u u_x; \quad u(0,x) = u_0(x)$ I have read that they can be solved exactly by the inverse scattering method, but ...
2
votes
0answers
103 views

Finite Difference Spacing of Points for PDE's for Convergence of Explicit Forward-Stepping Scheme

I realize that this question could be pretty broad, but I'm wondering at least what the conditions are for my simulation. I'm developing an Explicit Forward-Stepping Finite Difference scheme to solve ...
1
vote
0answers
13 views

Causality and viscous wave equation

I've seen several papers related to the causality condition concerning the viscous wave equation resolution but never understood how causality and stability are linked ? Conceptually, it seems hard to ...
1
vote
0answers
27 views

heat equation with Interface Crank Nicolson

I am currently working on solving the heat equation with an interface numerically using Crank-Nicolson. There are jump discontinuities at the interface which are dealt with using fictitious values ...
1
vote
0answers
32 views

Stability conditions

Below is a problem about stability conditions that I have been struggling with it during an exam: Find the stability conditions for $$A\left ( \frac{\partial^2 u(x,\, y,\,t)}{\partial x^2} + ...
0
votes
1answer
39 views

Finite Element method Implementation

I have written a program for the finite element method for an elliptic one dimensional problem.Initially I assumed a mesh that had only 10 points , but since the error was far above my tolerance ...
0
votes
1answer
93 views

numerical update rule for discretized hawkes excitation process

So I think I am just misunderstanding some simple notation or something and would appreciate some help. I am trying to replicate this model in an agent based model, but I cannot seem to figure out the ...
1
vote
0answers
62 views

Numerical solution of non-linear differential equation with MATLAB

I need some information to know if I can solve a nonlinear integral equation with terms $ u_{x} $ , $ u_{x}.u_{y} $ , $ u_{xx} $ , $ u_{xy} $ $u_{yy} $ $ u_{x}^{2} $ $ u_{y} ^{2} $ By numerical ...
0
votes
0answers
37 views

Errors in numericaly solving hyperbolic PDE in matlab

I am a beginner for PDE and I want to solve a hyperbolic PDE using matlab's builtin function hyperbolic(). However I am facing some erros and I could not resolve them. Can someone suggest or comment ...
4
votes
0answers
97 views

What differential equation might model this almost-harmonic oscillator?

I need to precisely control the motion of a damped, driven (nearly) harmonic oscillator: $$ \ddot x(t) + \alpha\dot x(t) + \omega_0^2 x(t) \approx V(t) $$ I use the $\approx$ symbol because this is ...
0
votes
0answers
27 views

Finite Differences and Scale Invariance

The semilinear heat equations $u_t = u_{xx} + u^p$ is invariant under the one parameter family of scalings $ t \to \lambda t, \quad x \to \lambda^{1/2}x, \quad u \to \lambda^{-1/(p-1)}u$. When we ...
0
votes
0answers
47 views

How to draw a diagram for the following PDE

Subject: Partial Differential Equations. Here are the details of the question: $$ \frac{\partial ^2u}{\partial x^2} + \frac{\partial ^2u}{\partial y^2} = 0 $$ for $0 < x < 3, 0 < y < 1$ ...
0
votes
2answers
61 views

finite difference scheme for nonlinear partial differential equations

I have the following second order partial differential equation (PDE) on $[0,T] \times \mathbb{R},~ T >0 $: \begin{equation} \left(1 + \frac{1}{(1 + b f)^2}\right) \frac{\partial f}{\partial t} ...
0
votes
0answers
33 views

Finite Difference Scheme for a PDE on non-rectilinear coordinates

Consider poisson's equation on the domain $0 \leq x \leq 1$ $0 \leq y \leq H(x)$. Change the coordinates to $\xi=x$, $\eta=y/H(x)$. Construct a FDS that gives a positive definite symmetric matrix. ...
2
votes
2answers
122 views

dual of $H^1_0$: $H^{-1}$ or $H_0^1$?

I have a problem related to dual of Sobolev space $H^1_0$. By definition, the dual of $H^1_0$ is $H^{-1}$, which contains $L^2$ as a subspace. However, from Riesz representation theorem, dual of a ...
0
votes
0answers
11 views

von Neumann analysis

When performing a von Neumann analysis on a PDE, we end up getting an expression in Fourier-space on the form (k is a "wave number", as in $\exp(ikr)$) $$ A_{n+1}(k) = G(k)A_{n}(k) $$ where $G$ is our ...
1
vote
2answers
64 views

Physical meaning of boundary conditions in the diffusion equation

I want to simulate the diffusion equation numerically. $ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$ With the boundary condition $\frac{\partial u}{\partial x} \bigg|_{x=R}=0 ...
1
vote
1answer
48 views

Advanced Topic in Numerical solution of Differential Equations?

i) IF $\frac{dy}{dt} = - \frac{∂H}{∂z}, \frac{dz}{dt}= \frac{∂H}{∂y}$ where H is a function of $y$ and $z$, show that $H(y,z)$ is constant in time. ii) Take a $H(y,z) =Ay^2 + 2Hyz + Bz^2$ where ...
0
votes
1answer
38 views

Advanced topic in numerical solution of differential equation

Investigate the stability of the PECE method where P=Predictor : $y_{n+1} = y_n + hf(y_n)$ C=Corrector: $y_{n+1} = y_n + h [(1-θ) f(y_n) + θ f(y_(n+1))], (0<θ<1) $ and E is the evaluation ...
0
votes
1answer
50 views

system of partial differential equation and boundary condition.

Let $\Omega$ be a regular domain, for example be a rectangle. Is it true that solve system of PDE's like this: $$u+\Delta w=0, ~~~~~~~~~~~~~~~~~~~~~~w=b1,~\frac{\partial w}{\partial ...
0
votes
0answers
49 views

two-point boundary value problem for elliptic equations (ODE)

we consider two-point boundary value problem $$Au=-au''+bu'+cu=f~~~~~~~~~~~~~~~~ in ~~\Omega=(0,1)$$ $$u(0)=u_0,u(1)=u_1$$ where $a=a(x)>0$, $b=b(x)\ne 0$ and $c=c(x) \ge 0$ We must prove ...
3
votes
0answers
78 views

Crank Nicolson Method PDE

I have the following PDE $0=\partial _t u+\frac{1}{2}\partial_{xx} u$, now I assume that $t\in[0,T], x\in[0,L]$ and initial data $u(T,x)=g(x), u(t,0)=a(t), u(t,L)=b(t)$ The grid $\{(ik,jh): ...
4
votes
0answers
33 views

How to approximate derivative inside derivative

I am using a box-scheme for solving partial differential equation. The function is approximated with: $$ f_p=\psi\cdot\left(\theta\cdot f_{j}^{k+1}+(1-\theta)\cdot f_{j}^{k} \right) + (1-\psi)\cdot ...
0
votes
1answer
52 views

Finite differences linear PDE

Suppose you want to write a finite difference scheme for a linear PDE of the form $ V_t + x V_x + x^2 V_{xx} =0 $, where the subscripts x,t denote derivatives with respect to space and time ...
1
vote
1answer
92 views

In the numerical solution of the Wave Equation, using finite differences, where do I obtain the spatial values from?

In trying to implement a simplistic numerical solver for wave equations, I have run into a conceptual problem that I haven't been able to solve. Consider a one-dimensional wave equation of a quantity ...
1
vote
1answer
19 views

Does the analytic solution of explicit scheme increase with time?

I would solve $U_t=U_{xx}$. To do that I do the approximation $U(x,t)\approx u_{p,q}$ and use the explicit scheme $$\frac{u_{p,q+1}-u_{p,q}}{\delta t}=\frac{u_{p-1,q}-2u_{p,q}+u_{p+1,q}}{(\delta ...
1
vote
0answers
52 views

Do PDEs arise in the context of Varieties?

My work involves designing numerical methods to solve PDEs on manifolds. This situation often arises in many applications in biology, chemistry, physics etc. I've recently been reading about ...
2
votes
1answer
40 views

How to add viscosity in speed function when calculating Fast Marching Method

Sorry, this's more like a question for programming rather than math. On implementing Fast Marching Method, I have a question for adding viscosity to the speed function so the evolved front cannot get ...
2
votes
0answers
31 views

Is numerical stability preserved under basis transformation for pde

I'm using a Backward Time Central Space method to solve the heat equation in polar coordinates. In Cartesian coordinates, it is easy to show this is unconditionally stable (by assuming solution of the ...
0
votes
1answer
22 views

von Neumann stability analysis for irregular meshes

All the litterature I have come across about the von Neumann stability analysis is performned on regular grids. Can the analysis be performed analytically on irregular grids, or does it have to be ...
0
votes
0answers
81 views

Discrete Harmonic Functions

Let $\Omega$ be a Lipschitz domain, $\Gamma\subset\partial\Omega$ with Lebesgue measure $>0$. Let $u_\Gamma\in V^h|_\Gamma$ be the trace of a finite element function. Then the following holds: $$ ...
3
votes
0answers
35 views

Finite Difference Methods for arbitrary elliptic PDE

I am looking for textbook references that describe lattice numerical methods for arbitrary elliptic PDEs, particularly finite difference schemes and particularly in 2d. The few references that I have ...
0
votes
1answer
51 views

zero curvature boundary value problem

Is it possible to solve a differential equation \begin{equation} f^{''}(x) = g(x) \end{equation} using finite difference method when the boundary condition is zero curvature (second derivative)? I ...
1
vote
1answer
46 views

FETI domain decomposition - kernel of local stiffness matrices

Consider the differential equation \begin{align*} -\Delta u&=f\mathrm{\ in\ }\Omega \\ u & = 0 \mathrm{\ on\ }\partial\Omega \end{align*} with $\Omega=(0,1)^2$. We're splitting $\Omega$ into ...
0
votes
1answer
42 views

Prove that $a(u-u_{h},u-u_{h})\ge 0$

Assume that $a$ is bilinear, symmetric and positive definite form, $u\in X$ and $u_{h}\in X_{h}\subset X$. I know the following fact: $$a(u-u_{h},u_{h})=0$$ Frm positive definiteness ...
3
votes
0answers
78 views

In which space does my (weak) solution exist?

I am reading through some numerical analysis papers, and was wondering what characteristics of a problem define which function space the weak solution of a problem belongs to. For example, for the ...
6
votes
1answer
247 views

Wave Equation Non-uniform string (PDE)

The wave equation in a non-uniform string is : $$ u_{tt} = c(x)^2 u_{xx} $$ $$ u(x,0) = f(x) = e^\frac{(x-\mu)^2}{2 \sigma^2} , \:\:u(0,t) = 0\:,\:\:u(L,t) = 0, \:\: u_{t}(x,0) = -cf'(x) $$ ...