Questions tagged [pde]

Partial differential equations (PDEs) are equations that relate the partial derivatives of a function of more than one variable. This tag is intended for questions on modeling phenomena with PDEs, solving PDEs, and other related aspects.

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13
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432 views

Sequential approach to solving coupled PDEs

I'm dealing with a coupled system of three transient, non-linear convection-diffusion equations. Let's just say to simplify the problem that they take the following form: $$ -\nabla\cdot(D_{1}(u_{2},...
9
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0answers
199 views

Time advance in Adaptive Mesh Refinement method

I am working on solving complex system of 2D PDEs governing the behaviour of plasma in a gas lamp during discharge. Recent tests have shown that because of steep gradients in temperature field and ...
9
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0answers
443 views

What's a good numerical/optimization software package for solving the 2-D optimal stopping problem?

I am looking for a numerical software package to help me solve the 2-dimensional "free boundary" PDEs that arise in optimal stopping problems. In one dimension a standard optimal stopping problem in ...
8
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0answers
88 views

How to construct an effective preconditioner for this particular problem

A quick introduction to my problem I am currently developing a method for simulation of water waves in three dimensions based on potential flow theory. The computational bottleneck of the method is ...
6
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0answers
161 views

Strategy for solving a non-trivial differential equation

I would like to numerically solve an equation of a type as shown below. Does anyone of you have an idea how to approach such a problem? Any links to literature or for further reading would be greatly ...
5
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0answers
60 views

Why does the naive barycentric hodgestar fail?

The discrete exterior calculus is defined first using circumcentric dual cells, because the primal and dual edges are orthogonal and thus the dual cells are convex. This leads to a diagonal hodge star ...
5
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90 views

Evolutionary dynamics in vascularised tumors, PDE-ODE coupled system

I have to solve the following PDE-ODE system $$ \displaystyle{\partial_{t} n = \bigl[a(s) - b(s)(y - h(s))^{2} - d\int_{\mathbb{R}} n \, dy \; + \; \beta \, \partial^2_{yy} n \;} \\\\ \...
5
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0answers
84 views

Numerical methods for the continuity equation with Sobolev vector field

Consider the continuity equation $$ \partial_t \rho(x,t) + \operatorname{div} (b(x,t) \rho(x,t)) = 0, \qquad t \in [0,T], \quad x \in \mathbb R^N, $$ with $b \in L^1((0,T), W^{1,p}(\mathbb R^N))$. ...
5
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0answers
507 views

Solving a PDE implicitly by iteration in python

Connected to this question here on Computational Science, I've posted a follow-up question on how to solve a PDE using an implicit scheme like Crank-Nicholson in general in this question on SO. But I ...
5
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0answers
96 views

Discrete sine and cosine transform for mixed derivatives

Using sine and cosine transforms to solve Poisson's equation with Dirichlet boundary conditions seem quite standard nowadays (see, e.g., here or Table 2 in this paper). In the case of Poisson's ...
5
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119 views

Using entropy functions for increasing numerical stability

Regarding the numerical stabilization of two-dimensional advection equation, \begin{equation} \dfrac{\partial f}{\partial t} + \Big(\dfrac{d\varepsilon_1(k)}{dk}\Big)\dfrac{\partial f}{\partial z} - \...
5
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332 views

Frozen coefficient method (von Neumann stability analysis)

Earlier it was considered that frozen coefficients method for Neumann stability analysis for finite difference scheme is more heuristic than rigorous. But I have read some information in a book by ...
5
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0answers
239 views

Negative viscosity stabilized by fourth order terms

I am trying to solve a "Navier-Stokes"-type problem where the viscosity is negative. Of course this renders the equation unstable and thus I add a fourth order term, so the entire equation becomes: $$...
5
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611 views

frozen coefficient vs. constant coefficient

This is a follow up to the question about the method of frozen coefficients. I think it deserves to be a separate question. The frozen coefficient problems are obtained by fixing the coefficients of ...
5
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172 views

Solving diffusion PDE using finite differences

I need some hints on how to solve this diffusion equation ($\alpha, k_1,k_2$ and $k_3$ are constants): $$ {\partial P \over \partial y} + k_1 {\partial P \over \partial t} + \alpha P = {1 \over k_2} ...
4
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0answers
174 views

How to apply an integrated constrain condition in FEM?

I'm running some simulation using FEM. In my model I need to apply a constraint condition to the governing equation. My governing equation similar to the diffusion equation as below: $$\frac{\...
4
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0answers
189 views

Eigenvalues of a Laplacian operator on an irregular mesh

I have the following setting: An irregularly-shaped domain, expressed as a mesh of points A Laplacian operator, together with boundary conditions I am looking for the eigenvalues of that operator, i....
4
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0answers
54 views

Phase dislocations and numerical accuracy

I am solving the nonlinear Schrodinger equation (NLSE), $$A_t+iA_{xx}+i|A|^2A=0$$ where $A$ is a complex valued function, which can be written as $A=ae^{i\theta}$ for $a,\theta$ real. Now, for ...
4
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149 views

numerical analysis of a partial integro-differential equation

I have to numerically solve a nonlinear partial integro-differential equation. This is my equation, $$\frac{\partial y(x,t)}{\partial t}=\int_{-1/2}^{1/2} \frac{\pi\cos u}{\sin\pi u-\sin\pi x} \frac{\...
4
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517 views

Fenics: Result of Steady state dynamic linear elastic doesn't match with actual values

I solved the steady state dynamic linear elastic model in a solid. My equation is a function of frequency and the strong form is: $$\operatorname{div}(\operatorname{stress}(\vec x, w)) + w^2 \rho u( \...
3
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0answers
66 views

Strange Picard iteration

I am interested in solving the equation $$ \begin{aligned} \nabla \cdot\left(\nabla \phi-\frac{\nabla \phi}{|\nabla \phi|}\right) &=0 & & \text { in } \Omega \\ \phi &=0 & & \...
3
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0answers
48 views

Solving PDEs: What is the best way to deal with non-banded/dense jacobians?

I have a system of PDEs describing atmospheric chemistry and transport. I use finite-differences to make my system of PDEs into a system of ~10,000 ODEs. I then integrate the ODEs forward in time with ...
3
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1answer
352 views

Poisson image blending artifacts

I am trying to implement Poisson image blending as in the paper Poisson Image Editing. This is the task of filling in a masked region of an image by minimizing $$\min_f\int_\Omega \left | \nabla f - \...
3
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0answers
57 views

Use of non-typical values of $\theta$ in theta-methods

The theta-method is a popular solution for solving time-transient PDEs (or ODEs), which consists of solving the general equation for each time step: $$ \frac{u^{n+1} - u^{n}}{\Delta t} + (\theta f(u^{...
3
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0answers
39 views

Guide for finite-difference schemes for Hamilton-Jacobi-Bellman Equations

I need to solve a simple, low-dimensional Hamilton-Jacobi-Bellman equation. Is there a simple guide for doing this numerically using finite-difference schemes? I found a few research articles ...
3
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0answers
109 views

How to account for the interface between two different phases in a discretized diffusion model?

I have tried to set up a model for the diffusion of a gas into a liquid. The two media are next to each other and the geometry is spherical because the system should simulate the diffusion out of a ...
3
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0answers
111 views

Piecewise Constant Enrichment of the continuous galerkin method

I am interested to study crack propagation in a hyperelastic material in a variational setting. The crack surface exhibits a jump discontinuity. The function space for displacement field should ...
3
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0answers
184 views

Multigrid for Robin boundary conditions

I have a question regarding the treatment of Robin boundary conditions in a multigrid solver. I am solving the Poisson equation in $\Omega=(0,1)^2$ with Robin boundary conditions on the boundary, $$- \...
3
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0answers
105 views

Spectral/hp-finite elements for 4th order PDEs

Does anyone know of references discussing the solution of 4th order PDEs by way of spectral/hp-finite element methods? Specifically, I'm interested in the extension of the spectral element method, ...
3
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0answers
191 views

Numerically solving a system of partial integro-differential equations in Matlab

Given the following system of partial integro-differential equations $$ \frac{dX(t)}{dt}=\Lambda-\mu X(t)-\beta X(t)Z(t),\\ \frac{\partial Y(t,\omega)}{\partial t}+\frac{\partial Y(t,\omega)}{\partial ...
3
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0answers
82 views

How should I choose the knot sequence when using B-splines as a basis for solving a PDE?

I'm looking to solve the Schrödinger equation with a basis made of a tensor product of basis splines. A number of papers describe calculations made with a program designed this way, but they never ...
3
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0answers
187 views

Integration of nonlinear PIDE via spectral methods

At the mean-field level, the dynamics of a polariton condensate can be described by a type of nonlinear Schrodinger equation (Gross-Pitaevskii-type), for a classical (complex-number) wavefunction $\...
3
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0answers
410 views

A Question About Weak Forms in Fenics

Is it possible to use test and trial functions from two different function spaces (defined over two different meshes) in a single weak form? Under what conditions can I do this (eg., each term in the ...
3
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0answers
345 views

Solving PDE or eigenvalue problems without FEM

Do you know any methods/solvers for PDE or eigenvalue problems like $\begin{cases} \Delta u= 0\ (\text{ or }\lambda u) & \text{ in }\Omega \\ u =0 & \text{ on }\partial \Omega \end{cases}$ (...
3
votes
1answer
208 views

Which discretization scheme to use for elliptic PDE?

While simulating motion of nonlinear inelastic wire one meets the following equations \begin{align} &{\partial^2\varphi\over\partial t^2}=2{\partial F\over\partial s}{\partial\varphi\over\partial ...
2
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0answers
61 views

Divergence on wave equation simulation

I'm currenly working on my own PDE solver for non-linear simulations in python. I've done succesfully simulations for KdV and Fisher's equation, but now I'm playing with second order derivatives in ...
2
votes
1answer
87 views

Developing a meshfree contouring algorithm

I would like to find the contours of a scalar function $u(x,y)$ available as a discrete set of function values $u_i = u(x_i,y_i)$ over a scattered set of points $(x_i,y_i), i=1,...,N$. In my case, the ...
2
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0answers
41 views

In sights into why higher order finite differencing method leads faster to instability

I was playing around with numerically solving the 1D wave equation with density and stiffness varying with position using central differencing methods and noticed that for certain discretization steps ...
2
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0answers
84 views

What are the practical differences between a Galerkin finite element method vs an orthogonal collocation method?

What are the practical differences between a Galerkin finite element method vs an orthogonal collocation method? The best reference I've found thus far as been a paper titled The Performance of the ...
2
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0answers
40 views

How to increase the stability of a DAE solver?

I am trying to solve a set of linear PDEs of the form $$F\left(\vec{y},\frac{\partial \vec{y}}{\partial x},\frac{\partial^2 \vec{y}}{\partial x^2},\frac{\partial \vec{y}}{\partial t}\right)=0.$$ To ...
2
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0answers
27 views

Multidimensional center-of-mass coordinates for PDE solutions

In this question I'm looking for a discussion on coordinate transformations. Say I have a PDEs of the form $$ \frac{\partial}{\partial t} \, V(t,x) \ = \ \Bigg\{r(x,t) - \sum_{m=1}^D \mu_m(t,x)\frac{\...
2
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0answers
65 views

How to determine the finite difference coefficient matrix in 2D with periodic BC?

I'm solving a PDE in matlab using ode15s, and since the spatial dimension is 2, and number of variables grow large very quickly, I need to supply the structure of ...
2
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0answers
243 views

Extracting FEM matrices in matlab pde toolbox

I am trying to follow the dynamic linear elasticity in Matlab, link here. My goal is to extract the FE Matrices using the function assembleFEMatrices in matlab and solve the resulting system of second-...
2
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0answers
88 views

Partial differential equation with convolution integral

The GP equation is $$i\frac{\partial u}{\partial z}+\nabla^2u+|u|^2u+\int e^{-[(x-x')^2+(y-y')^2]}|u(x',y')|^2 dx'dy'u(x,y)=0$$ with Neumann boundary condition. The initial condition is given by a ...
2
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0answers
82 views

How to solve $y(x) y'''(x)=f(x)$

I have a PDE of the form $\partial_t y(x,t)+\partial_x(y(x) y'''(x)-f(x))=0$, where $f(x)=\cos(x)$. Suppose a stable equilibrium exists, and I want to find the steady-state solution $y(x) y'''(x)=f(x)...
2
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0answers
43 views

How to solve the Poisson equation with KINK aligned with mesh facet

I have a problem that solving the Poisson equation with kink ( discontinuous gradient but solution is continuous ) in the analytical solution, I want to solve this problem with FEM. To approximate ...
2
votes
0answers
83 views

Identify the components of the (weak form) PDE in structural mechanics

I am trying to identify the weak form of PDE in structural mechanics. I read a lot of papers where they are using the elliptic boundary value problem \begin{equation} \int\limits_{\Omega} \delta \...
2
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0answers
129 views

Neumann boundary conditions in the Maccormack scheme

I am trying to solve the viscous Burger equation $$ \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} = \mu \frac{\partial^2 u}{\partial x^2} $$ with Neumann boundary conditions. I am ...
2
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0answers
156 views

Analysis and numerical methods for PDEs arising in industrial problems

Assume some basic knowledge of numerical methods for PDEs, acquired through A. Quarteroni's Numerical models for differential problems. I'm looking for a reference to get started on the analysis of ...
2
votes
0answers
136 views

What is Chebfun `eigs` doing

What is this doing? Looks like the original eigenvalue problem is converted into generalized eigenvalue problems with different dimensions of collocation points. Can someone explain more about this? ...