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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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 ...
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42 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 $u_d\in H^{1/2}(\partial\Omega) \times \mathbb{R}^+$. ...
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42 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 non-negative everywhere in the domain. Since I am working in fourier space for solving this problem ...
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1answer
73 views

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

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: ...
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11 views

Constraint condition in comsol [closed]

In Comsol I am working with general form of PDE. I have made a rectangular geometry and solving equation for that. In this there are two condition as Initial value and zero flux are in-built. I want ...
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1answer
60 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 ...
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2answers
126 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 ...
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33 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} - ...
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1answer
112 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 ...
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24 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 ...
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1answer
61 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 ...
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31 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 ...
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2answers
105 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}} ...
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1answer
37 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 ...
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41 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 ...
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1answer
102 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 ...
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1answer
112 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 ...
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51 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 ...
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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 ...
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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) + ...
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1answer
63 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 ...
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1answer
94 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: $$ ...
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90 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 ...
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3answers
96 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} = ...
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33 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 ...
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1answer
99 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 ...
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1answer
144 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 ...
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1answer
93 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), ...
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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 ...
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3answers
159 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 ...
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2answers
190 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 ...
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1answer
118 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 $N$ ...
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60 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}_+ ...
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84 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 ...
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1answer
144 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 ...
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1answer
80 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, ...
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1answer
135 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 ...
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97 views

1 D Diffusion equation FDM with different layers

I'm trying to solve this particular equation $\frac{\partial u}{\partial t} = \frac{\partial}{\partial x} \big[D_{i}(x)\frac{\partial u}{\partial x} \big] + S(x,t)$ where the $i$ index denotes ...
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1answer
52 views

Reaction-Diffusion problem A->B, solving for B

I need to solve a Reaction-Diffusion using Finite Elements, piecewise linear elements. In this problem, a reaction $A \rightarrow B$, with rate law $ r_A = - k_A \cdot u_A $, takes part, where $u_i$ ...
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33 views

Isolated solving of coupled system of PDE [duplicate]

I would like to solve 3 differential equations for 3 unknowns. So I wrote a MATLAB code, which solves (using the '\' operator) these equation using a linear system of equations (in which the 3 ...
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2answers
145 views

Usability of upwind finite difference schemes

NOTE: I asked this on Mathematics Stack Exchange and there were no answers. So, I thought I might try here. Upwind schemes like the classic "upstream" scheme, can be used to solve, for example, the ...
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2answers
149 views

How do hexahedral FEM meshes improve approximation quality per degree of freedom, compared to tetrahredal meshes?

From the deal.II FAQ : ...quadrilaterals and hexahedra typically provide a significantly better approximation quality than triangular meshes with the same number of degrees of freedom; you ...
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0answers
89 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_{-\infty}^\infty K_0(|x-u|) \frac{\partial^2 ...
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3answers
173 views

Finite elements on manifold

I'd like to solve some PDEs on manifolds, say for example an elliptic equation on a sphere. Where do I start? I'd like to find something that use preexisting code/libraries in 2d , nothing so fancy ...
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1answer
168 views

Why are functional representations of systems important in numerical applications?

I tried asking a similar question in SE.Physics, and I got some information regarding the abstract side of this, but I figured I should post here to get more complete information about the numerical ...
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2answers
142 views

Well-posedness of a linear elasticity problem and Navier-Cauchy equation

I read a master thesis on a topic I'm interested too. This work concern the solution of the displacement equation of motion for a homogeneous, elastic, isotropic material: $$\rho \ddot{\mathbf{u}} - ...
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1answer
91 views

What does “strongly conservative” mean in the context of numerical methods?

I have a homework problem that asks me to show that 1st order unwinding or central differencing can give a strongly conservative, consistent scheme for the 1-D Burger's Equation using a finite volume ...
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3answers
152 views

Algorithms for radiation treatment planning

I have a medical physics problem - I want to maximise the dose absorbed by a brain tumour whilst minimising the dose in the rest of the brain, especially certain organs, such as the pituitary gland, ...
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3answers
133 views

Variable viscosity Stokes equation

One very efficient way to solve Stokes equation with periodic boundary conditions \begin{equation} -\eta \nabla^{2} \bf{v} + \nabla p = f \\ \nabla \cdot \bf{v} = 0 \end{equation} is using the ...
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164 views

Stationary 2D/3D Navier-Stokes source code

Trying to solve stationary Navier-Stokes problem for incompressible laminar Newtonian fluid. I've found a couple solutions for instationary Navier-Stokes equations (like FeniCS examples or CFD ...