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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37 views

Numerical methods for solving a hyperbolic nonlinear PDE

What type of numerical methods are there to solve PDE of the sorts of: $$\begin{align} &f(x,t,u(x,t))u_{xx} - g(x,t,u(x,t))u_{tt} = F(x,t,u(x,t))\\ &u(x,0)=G_1(x)\\ &\frac{\partial ...
0
votes
0answers
27 views

Semi-discrete advection equation and plot [on hold]

So, I'm supposed to do a semi-discrete scheme of the advection equation with periodic b.c. using central differences of 4th order. $u_x + u_t = 0$, $0 < x < 2pi$ $u(0,t)=u(2pi,t)$, ...
1
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0answers
60 views

Discontinuity at Interface

The equation at the left of the interface is \begin{equation} \displaystyle\frac{\partial C_i}{\partial t} = D_i \nabla^2 C_i - z_i \frac{D_i}{RT}F \nabla \cdot (C_i \nabla \phi_2) \end{equation} ...
1
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0answers
38 views

Jacobi method converging then diverging

I am working to solve Poisson's equation in 2D axisymmetric cylindrical coordinates using the Jacobi method. The $L^2$ norm decreases from $\sim 10^3$ on the first iteration (I have a really bad ...
5
votes
1answer
51 views

Can Variational Inequalities handle non-symmetric matrices?

I am trying to enforce the discrete maximum principle (i.e., ensuring non-negative concentrations) for diffusion-type problems that have an anisotropic diffusivity tensor (e.g., tensor dispersion from ...
3
votes
0answers
68 views

What should be the number of boundary conditions of a PDE

As far as I know, for getting a unique solution to a PDE we should impose some boundary conditions to the PDE. "The number of required auxiliary conditions is determined by the highest order ...
0
votes
1answer
50 views

adjoint method package for ODE(PDE)-constrained optimization

I have this type of question (ODE-constrained optimization) to solve: $g(x,p)=0$ is the simulation, where $x$ is state variable and $p$ is parameters aimed to optimize; $f(x)$ is the objective ...
2
votes
2answers
95 views

Ill-conditioned Jacobian matrix from Nernst-Planck equation with Butler-Volmer reactions

The governing equations are listed here of my notes on page 4. It's a reproduction of other's paper which solves the equations with COMSOL. The problems arise when I want to solve for the consistent ...
1
vote
2answers
76 views

Building minimization optimization problem for 2nd-order elliptic PDE

I am solving elliptic PDE problem, for which, Euler scheme looks as following: $$ \nabla [\gamma ( |\nabla u|^2) \nabla u] = 0,$$ where $$\gamma(|\nabla u|^2) = (1 + |\nabla u|^2)^{-1/2}. $$ I am ...
4
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0answers
35 views

Flux at coarse-fine mesh grid interface?

I am trying to solve one dimensional inviscid Burger's equation using adaptive mesh refinement. This is the PDE: $$\frac{\delta U}{\delta t} + \frac{\delta F}{\delta x} = 0$$ where the flux F of the ...
0
votes
1answer
101 views

Why is Godunov's scheme (for the advection equation) diffusive?

I'm trying to solve the advection equation $$m_t+(\alpha m)_x=0$$ with $m(0,\cdot)=m_0$ numerically using the first order Godunov scheme. Hence I write ...
3
votes
1answer
69 views

Computing Fourier representation of space dependent advection operator via FFT

Consider the following equation on the circle: $$\dfrac{\partial p(x,t)}{\partial t} = a(x)\dfrac{\partial p(x,t)}{\partial x} \equiv L(p) \enspace ,$$ where $L$ is the operator acting on $p(x,t)$. ...
1
vote
1answer
27 views

Spatial evolution of kinetic energy in free surface flow in terms of values on boundary

I'm running a potential flow solver, and I have values of the velocity potential $\phi$ on the boundaries. I'd like to compute the spatial evolution kinetic energy in the system. In particular, I'd ...
1
vote
1answer
70 views

I need to scale variables to solve a 2D PDE. What are the physical considerations of scaling?

I am solving a boundary value problem in 2D via an implicit finite difference scheme. Unfortunately, although the problem is well-posed and should have a unique solution, the condition number of the ...
1
vote
1answer
65 views

Computation of plane wave scattering on semi infinite plane

I have attempted to code up the simple math required to plot the total field set up by an incident plane wave on a semi-infinite flat plate which can be found here. To summarise: $$\phi_s(r,\theta ) ...
4
votes
1answer
71 views

Comparing finite differences methods

I am currently writing my dissertation on different methods for pricing barrier options. As part of this, I have implemented a finite differences method for solving one partial differential equation, ...
4
votes
2answers
130 views

How to discretize Burger's equation?

I am trying to solve the very simple one dimensional burgers equation which is: $$\frac{\delta U}{\delta t} + \frac{\delta F}{\delta x} = 0$$ where the flux F of some variable U is defined as$$ F= ...
0
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0answers
33 views

Non-overlaping Domain decomposition - assemble of Laplacian

I am dealing with following 2-dimensional problem in the unit square domain $S_2$ $$- \Delta u (x,y) = f \ \text{in} \ S_2, \hspace{1.5cm} u(x,y) = 0 \ \text{on} \ \partial S_2$$ where $f$ is ...
7
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0answers
104 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
38 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 ...
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0answers
60 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
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2answers
66 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
47 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
124 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
113 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
114 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 ...
6
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0answers
127 views

Verification in Eigenvalue problems

Let us start with a problem of the form $$(\mathcal{L} + k^2) u=0$$ with a set of given boundary conditions (Dirichlet, Neumann, Robin, Periodic, Bloch-Periodic). This corresponds with finding the ...
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0answers
32 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
66 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
98 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
69 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
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0answers
60 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
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1answer
94 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
153 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 ...
1
vote
1answer
119 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
80 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
139 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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0answers
39 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
119 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
32 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
65 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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0answers
34 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
108 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
40 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
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0answers
49 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
122 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
125 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
56 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
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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 ...