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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6
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0answers
36 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 ...
2
votes
0answers
17 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 ...
0
votes
0answers
56 views

Solving quasilinear/nonlinear equations obtained from the discretization of partial differential equations

When you solve numerically a (system of) linear partial differential equation (PDE) as for example Lapace's equation $\nabla^2\varphi = 0$ or Poisson's equation $\nabla^2\varphi = f$ you obtain a ...
2
votes
2answers
49 views

Scipy OdeInt solver with Neumann boundary conditions

I'm using scipy.odeint to solve Fisher-Kolmogorov equation: \begin{equation} u_t = u_{xx}+u(1-u) \end{equation} The code can be found here. From Ablowitz and ...
1
vote
1answer
34 views

Raviart-Thomas elements global definition and compact support

As per the suggestion by Christian in the comments here, as part of my continuing quest to understand the Raviart-Thomas (RT) elements I'd like to know how exactly the RT elements are defined ...
2
votes
1answer
81 views

Finite difference for nonlinear system of equation

\begin{equation} \frac{\partial C_i}{\partial t} = D_i \nabla^2 C_i - \frac{I \cdot \nabla t_i}{z_i F} - \sum_{i'} \frac{z_{i'}}{z_i} D_{i'}\nabla \cdot (t_i\nabla C_{i'}) \end{equation} ...
5
votes
1answer
95 views

Raviart-Thomas elements on reference square

I'd like to learn how the Raviart-Thomas (RT) element works. To that end I'd like to analytically describe how the basis functions look on the reference square. The goal here is not to implement it ...
2
votes
1answer
112 views

Darcy flow finite elements

The Darcy equations for porous media flow are given by: $\frac{\mu}{\kappa}\mathbf{u} - \nabla p = \mathbf{0}$ $\nabla\cdot\mathbf{u} = 0$ where $\kappa$ is the permeability and can in general be ...
5
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0answers
89 views

Verification in Eigenvalue problems

Let us start with a problem of the form $$(\mathcal{L}u + k^2) u=0$$ with a set of given boundary conditions (Dirichlet, Neumann, Robin, Periodic, Bloch-Periodic). This corresponds with finding the ...
1
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0answers
28 views

Monotonic convergence of Newton's method for boundary value problems

I’m interested in solving nonlinear elliptic boundary value problems of the type $$ -a\Delta u + f(u) = 0, $$ $$ u|_\Gamma = u_0 $$ by Newton’s method when its convergence is global and monotonic. ...
3
votes
1answer
65 views

How to get Fourier transform of Fisher-Kolmogorov?

How can I use Fourier Transform to solve Fisher-Kolmogorov Equation in 1D? \begin{equation} u_t(x,t) = u_{xx}(t) + u(1-u) \end{equation} \begin{equation} u(0,x) = \phi(x) \end{equation} with ...
1
vote
1answer
150 views

$AX=B$: How to solve for $X$ if elements of matrix A are matrices

Objective: I am trying to solve for $C$ in 2D space (x,y) and time from following PDE. $$ \text{PDE: }\frac{\partial C}{\partial t} + \nabla\left(v.C - D\nabla{C} \right)= \alpha.C $$ Method: I ...
4
votes
2answers
94 views

Periodic boundary condition for the heat equation in ]0,1[

Let us consider a smooth initial condition and the heat equation in one dimension : $$ \partial_t u = \partial_{xx} u$$ in the open interval $]0,1[$, and let us assume that we want to solve it ...
4
votes
1answer
59 views

Method of Manufactured Solutions for non-differentiable coefficients

The Method of manufactured is commonly used for verification of computational science codes. I want to use the method for verification of Navier-Cauchy (elasticity) equations with periodic and ...
1
vote
0answers
54 views

Solving a system of 4 coupled PDEs representing variable diffusivity

I have four partial differential equations representing mass conservation of two compressible fluid phases (marked by subscripts $p1$ and $p2$) in two different continuum media (marked by subscripts ...
2
votes
1answer
85 views

Bounded Input Boundaed Output stability for heat equation. Proof or Counter example?

I am interested in proving or obtaining a counterexample to the following conjecture. Let $\Omega\in \mathbb{R}^d$ be a bounded open domain. Let $u_d\in H^{1/2}(\partial\Omega) \times \mathbb{R}^+$. ...
7
votes
2answers
150 views

Enforcing non-negative constraint in fourier-spectral method

I have a PDE optimization problem, and a scalar field (which I am optimizing over) is supposed to be nonnegative everywhere in the domain. Since I am working in Fourier space for solving this problem ...
0
votes
1answer
98 views

Implementation of 1D Advection in Python using WENO and ENO schemes [closed]

I'm trying to implement 1D advection solver using WENO and ENO schemes. \begin{equation} \frac{\partial u}{\partial t} + \frac{\partial f(u)}{\partial x} =0 \end{equation} where: ...
2
votes
1answer
75 views

What are acceptable boundary conditions for porous media flow?

I am attempting to simulate fluid flow through a porous foam. I would like to have no-slip boundary conditions on part of the boundary and free flow conditions on the inlet and outlet. Right now I am ...
3
votes
2answers
137 views

1D inhomogeneous Poisson PDE with Dirichlet BCs, slow convergence

For an assignment I have to implement a 1D Poisson PDE with inhomogeneous Dirichlet BC's $$\Delta_1 u = f, \quad u(a)=g(a), \: u(b) = g(b) $$ I have managed to make it work, but I am not seeing the ...
1
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0answers
38 views

Nonlinear 2D modeling of Neural Electromagnetic field in Matlab

I am trying to replicate the MATLAB simulation presented in this paper. More specifically, I have to code the solution to this equation $$\frac{a}{2R_i}\frac{\partial^2 V_m}{\partial x^2} - ...
3
votes
1answer
117 views

Challenges in implementing Algebraic Multigrid on millions of processors

I just implemented an Algebraic Multigrid solver for a Mixed Dirichlet-Neumann Boundary Value problem and was surprised to see the speed-up as compared to a simple iterative solver for a large problem ...
0
votes
0answers
27 views

Matlab pde toolbox manipulate FEMesh object

Is there a possibility to write data to a FEMesh object? What I am looking for is an analogue to the MeshToPet method, something like a PetToMesh function. Any other way to manipulate the nodes in a ...
1
vote
1answer
63 views

A doubt in Multigrid V-cycle

Assume I have 3 levels of grids. Finest Grid = level 2, Coarser Grid = level 1, Coarsest Grid = level 0. Relax $u$ on $Au = b$ at level 2 for 3 times. Find residual $r2$ at level 2, then restrict to ...
0
votes
0answers
33 views

Compute solution of a pde with multiple boundary conditions

What are some general methods which can allow to solve the equation $-\Delta u = 0$ on a two dimensional domain, with mixed boundary conditions? There are a few methods I have in mind: finite ...
0
votes
2answers
107 views

Relationship between FEM solutions of PDE with different spatial resolutions

I use FEM to simulate deformations of elastic objects for animation applications in computer graphics. The governing equation is generally with the form: $$ \mathbf{M}\ddot{\mathbf{u}} ...
2
votes
1answer
38 views

Finding the frequencies of vibration of a circular and square drum

I want to find the frequencies of vibration of a circular and square drum. To do this, I need to solve a 2-dimensional wave equation (PDE) with boundary conditions. Every method that I have researched ...
1
vote
0answers
46 views

scipy.integrate.ode ignores boundary conditions

I am trying to solve the 1-dimensional diffusion problem numerically using method of lines: $$ \frac{\partial c}{\partial t} =D \frac{\partial^2 c}{\partial z^2},$$ where the right hand side is ...
2
votes
1answer
107 views

Computational methods for finding the energy eigenvalues of the time-independent Schrodinger equation with arbitrary potential

I have seen in some papers that the energy levels in some arbitrary potential are specified. How can one find the energy levels in such arbitrary potentials. For example, $V(x)=\sin^2(x/2)$ with ...
10
votes
1answer
118 views

Can an approximated Jacobian with finite differences cause instability in the Newton method?

I have implemented a backward-Euler solver in python 3 (using numpy). For my own convenience and as an exercise, I also wrote a small function that computes a finite difference approximation of the ...
2
votes
0answers
54 views

Numerical integration when solving PDE: Simpsons rule and high frequency noise

I am solving a PDE, and one of the intermediate steps is to numerically integrate a function over a compact interval. The function is represented on a linearly spaced grid. I am using Simpsons rule ...
0
votes
1answer
60 views

How can PDE matrices be identified?

I need to include experimental results for lots of PDE (partial differential equation) matrices in my research work. How can I identify PDE matrices? For example, matrices in the UFL Sparse Matrix ...
0
votes
0answers
53 views

The centered difference operator for fractional function

Recently, I come to a question about the 2nd order centered finite difference approximation of a fractional function, more precisely, we set $\delta^{2}_x u(x,t) = u(x_{i-1},t)-2u(x_i,t) + ...
4
votes
1answer
64 views

How can I compare errors in PDE solvers with non-uniform grids?

Is there a standard approach to testing codes with refined regions? Specifically, I am interested in testing whether the refinement is working correctly. For the sake of simplicity, let's consider a ...
0
votes
1answer
104 views

How to impose Neumann boundary conditions in interior penalty DG method

Consider the following two point BVP: $$ -u''(x)=f(x),~~~u(0)=u(1)=0. $$ An interior penalty DG method for this BVP that weakly imposes homogeneous Dirichlet boundary conditions is of the form: $$ ...
1
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0answers
93 views

Tips on improving stability in numerical scheme for non-linear PDE

I am solving a non-linear second order system of PDEs in two variables. The equations are too complicated to write out here, but an essential feature is that there is a propagating wave which then ...
6
votes
3answers
105 views

Stability criterion for waves in anisotropic solids

The equations of motion for an elastic solid are given by $$\begin{align} &\nabla \cdot \boldsymbol{\sigma} + \mathbf{f} = \rho \ddot{\mathbf{u}}\\ &\boldsymbol{\sigma} = ...
0
votes
0answers
34 views

Matlab PDE toolbox

I'm new at matlab and I can't figure out how to make external temperature not constant but as a function. I wanted to use pde toolbox, I choose parabolic type. My external temperature should be ...
1
vote
1answer
124 views

Instability of pdepe in Matlab… boundary conditions?

here is a Matlab beginner banging his head on the wall... I am trying to solve a system of partial differential equations in Matlab, with both derivatives in time and space domains. I am using the ...
3
votes
1answer
146 views

Boundary conditions in conforming Galerkin method for biharmonic equation

I am trying to solve simple scalar biharmonic equation using bubnov-galerkin finite element method. I am using $H^2$ conforming basis functions. I was wondering that if anyone can give me some ...
0
votes
1answer
99 views

Solving a PDE using Matlab (with varying initial conditions)

I want to solve a 1-D heat conduction PDE using Matlab which looks like $$ \rho c_p \dfrac{\partial T}{\partial t} = \dfrac{\partial}{\partial z}\left( \lambda \dfrac{\partial T}{\partial z} \right), ...
0
votes
2answers
86 views

Explain this multivariate differential identity

$$ \frac{\partial|\nabla\phi|^2}{\partial\phi}=-2\nabla\cdot\nabla\phi$$ I would very appreciate that you help me . Please do it in detail, I am quite not good at such problems. There is something ...
4
votes
3answers
168 views

Solving Laplace's equation on a domain with moving boundary

Consider a function $X(\xi,\nu)$, $2\pi$ periodic in $\xi$ satisfying $$\nabla^2 X = 0$$ in a domain $D$ with $\nabla = (\partial_{\xi},\partial_{\nu})$. If I know the values of $X$ on the boundary ...
1
vote
2answers
200 views

Python, numpy and complex functions (PDE's)

Update 4 I have almost given up on getting this right. This is the solution to the time-independent Schrodinger's equation, so the analytical solution is: $\psi(x,t) = \psi(x,0)e^{\frac{-iE ...
0
votes
1answer
133 views

Solving PDE with state and time dependent boundary conditions

I am interested in solving the following PDE (heat equation): $$\frac{\partial u}{\partial t} = \kappa \frac{\partial ^2 u}{\partial x^2}$$ In order to solve it, I discretize space uniformly into $N$ ...
0
votes
0answers
64 views

Solution to PDE with differential boundary conditions

I have the following equations $$ a_t(x,t)=1-a(x,t)b(x,t)^\gamma+D_1a_{xx}(x,t) $$ and $$ b_t(x,t)=\alpha(a(x,t)b(x,t)^\gamma -b(x,t))+D_2b(x,t)_xx $$ where $a,b:]0;4\pi[\times \mathrm{R}_+ ...
0
votes
0answers
87 views

parallel computatioan of a PDE in MATLAB

I want to solve a 1-D PDE $(\partial_{tt} + \alpha\partial_t)u(x,t)=\partial_{xx}u(x,t)-\sin(u(x,t))+f$, using method of lines and for this I defined a spatial grid of about n~1000 points. Since my ...
6
votes
1answer
161 views

How to project a vector into the H(div) space (in the context of finite elements)?

Say I have a simple elliptic PDE: $$ -\nabla\cdot(K\nabla p) = f \;\;\;\text{in}\;\Omega $$ with the appropriate boundary conditions. I solve for $p$ using a FEM (a discontinuous Galerkin method to ...
6
votes
1answer
99 views

Space-time finite element discretization for time-dependent PDEs

In FEM literature, semi-variational methods are typically used in the solution of time-dependent PDEs. I have not seen a fully-variational approach i.e. where space and time are discretised by FEM, ...
8
votes
1answer
136 views

Can the method of lines be used to discretize all PDEs?

I have found that the method of lines is a very natural way to think about the discretization of PDE's. Therefore I always default to that mindset when presented with a new set of equations. I have ...