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.

learn more… | top users | synonyms

0
votes
1answer
33 views

How can we compute the exact value of some rotational symmetric n-by-n convolution kernel

The commonly used $3\times 3$ Laplacian convolution kernel $\frac{1}{6}\begin{bmatrix}1 & 4 & 1\\4 & -20 & 4\\1 & 4 & 1\end{bmatrix}$ is only an approximation of rotational ...
2
votes
2answers
82 views

Can the conservative form of the advection equation be re-written by replacing the velocity term with an integral over all other points in space?

Suppose I have a 1D advection equation in conservation (divergence) form $\partial_t u(x,t) = -\partial_x [v(x)u(x,t)],$ where $u$ is a conserved quantity in space, and $v$ gives the velocity of the ...
3
votes
1answer
85 views

Inverse advection-diffusion problem, solving for a drift coefficient with experimental data?

I am investigating a physical process where I believe the 1-D advection-diffusion equation: \begin{equation} \frac{\partial u}{\partial t} = -\frac{\partial}{\partial x}[\mu(x,t) u(x,t)] + ...
1
vote
0answers
36 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 ...
1
vote
1answer
103 views

What kind of test cases are convenient to use for testing the code for Euler equations of gas dynamics in polar coordinates?

Consider Euler equation of gas dynamics in polar coordinates as $$ \left( \begin{array}{ccc} \rho \\ \rho u_{r} \\ \rho u_{\theta} \\ E \end{array} \right)_{t} + \frac{1}{r} \left( ...
1
vote
0answers
55 views

MATLAB: incorrect computation with pdepe for parabolic-elliptic system

I have found that MATLAB solves 1D parabolic-elliptic system incorrectly by using pdepe function. Here is a system: $$ u_t = u_{xx} + 2, $$ $$ 0 = v_{xx} + u. $$ Boundary conditions: $$ ...
5
votes
1answer
120 views

Numerical method for solving PDE with non-linear boundary conditions

I have the following problem: \begin{align} \frac{\partial w}{\partial t} = \frac{1}{r^2} \frac{\partial}{\partial r}\left( r^2 D_1 \frac{\partial w}{\partial r} \right) \\ \\ \frac{dr_d}{dt} = ...
0
votes
0answers
57 views

Eigenvalue analysis of preconditioned partial differential operator

today, I encountered a confused problem by accident, but I have no ideas to deal with it fully. The question can be described as follows: for example, when we need to use FDM/FEM to discrete the ...
2
votes
2answers
70 views

Infinite Function Value on Dirichlet Boundary

I have been working on a multigrid solution to a non-homogeneous Dirichlet boundary value problem. However, the function goes to infinity on the boundary. This causes numerical overflow errors to be ...
2
votes
0answers
49 views

4th order tensor [closed]

I'm new with FEniCS and Python and I'm stuck with this issue: is there a way to write a 4th order tensor in an easy way to implement? I have to compute the following stiffnes tensor: $A_{ijkl}= ...
1
vote
1answer
48 views

What are the good testing problems for hyperbolic equation?

I read the whole list of this question: Where can one obtain good data sets/test problems for testing algorithms/routines? But the answers are in different areas and I want to ask a specific area. I ...
3
votes
2answers
64 views

Adaptive mesh refinement algorithms and the difference between AMR and moving mesh

I'm working on my thesis and a part of it has to do with adaptive mesh refinement. As a computer science major, I'm not too familiar with this field. The best way I can put my knowledge of AMR is: I ...
0
votes
0answers
39 views

Solvers for nonlinear parabolic PDEs

Could you please advise some programs or libraries for solving parabolic PDEs (or its systems) in 1D, 2D and 3D, for example, with the method of lines? The system of parabolic PDEs can be nonlinear in ...
2
votes
0answers
50 views

Solvers for stiff initial value ODEs with sparse Jacobian

What ODE solvers are optimized for solving stiff systems with sparse Jacobian? Such systems appear, for instance, when a parabolic PDE is discretized in space using typical finite difference or finite ...
1
vote
2answers
188 views

How can I solve this 1D nonlinear, variable-coefficient hyperbolic PDE?

I need to solve the following hyperbolic equation in x and phi co-ordinates $$\frac{\partial \left ( -s/f \right )}{\partial \varphi }+\frac{\partial \left ( 1/f \right )}{\partial x}=0$$ $$\varphi ...
4
votes
2answers
223 views

Crank-Nicolson method for solving nonlinear parabolic PDEs

Is the Crank-Nicolson method appropriate for solving a system of nonlinear parabolic PDEs like $\partial u/\partial t - a\Delta u + u^4 = 0$ ? I tried to apply this method for solving such system but ...
-1
votes
1answer
46 views

Libraries with the method of lines for parabolic PDEs [closed]

Could you please advise some programs or libraries for solving parabolic PDEs (or its systems) in 1D, 2D and 3D, for example, with the method of lines? The system of parabolic PDEs can be nonlinear in ...
3
votes
2answers
115 views

How to set up a shock tube problem such that the solution includes a shock with a specified Mach number

One of the famous and convenient test cases for shock wave modeling is the 1D Sod's shock tube. This is a Riemann problem for the compressible Euler equations of gas dynamics. The initial set up has ...
0
votes
2answers
134 views

Numerical method of lines for solving PDEs

Could you please advise some literature about the numerical method of lines (MOL) for parabolic PDEs? It is a method of solving PDEs with discretizing only by space but not by time. A system of ODEs ...
1
vote
1answer
135 views

Solving Advection (Convection) - Diffusion - Reaction Partial Differential Equation in Python

I am looking for library written in Python which will enable me to solve the coupled nonlinear equations which looks like: I need the library which will enable me to couple this solver to other ...
2
votes
1answer
90 views

What is the exact formulation of compressible Euler equation of gas dynamics in polar coordinates with artificial diffusion in 2D?

The interested equation is advection-diffusion equation. One of the canonical example is Navier-Stokes equations. However, I would like to let the coefficient of diffusion constant goes to zero, ...
1
vote
2answers
61 views

Looking for some literature about numerical solution of _coupled_ PDEs

There's a vast amount of literature about the numerical solution of partial differential equations. But in none of my books or lectures did I find anything about coupled systems. Internet and library ...
3
votes
1answer
101 views

steady state solution from parabolic problem vs solution of elliptic problem

My question is related, but not a duplicate of Asymptotic convergence of the solution to a parabolic pde to the solution of an elliptic pde Suppose I solve the parabolic PDE: $u_t = \Delta u + ...
5
votes
2answers
157 views

Implementation of nonlinear term in FEM

Although there are similar questions, I am also struggling with the implementation of the following term in "my own code" by Finite Element Method, namely, $\nabla \phi \cdot \nabla \phi$. $\phi$ is ...
3
votes
1answer
94 views

Numerically solve a PDE in Python with a term calculated by coarse-graining

I'm trying to solve a PDE in Python of the form, $\dfrac{\partial c(\mathbf{x}, t)}{\partial t} = \mathrm{D} \nabla^2 c(\mathbf{x}, t) -\gamma \rho(\mathbf{x}, t) c(\mathbf{x}, t)$ where $c$ ...
3
votes
1answer
107 views

Partial differential equations with octave [closed]

I need to find a numerical solution for $-\Delta U = f$, on the $\Omega = [0,1]^2$, with $ U|_{\partial \Omega} = 0$. I found a method: POISSONFD in ...
5
votes
1answer
120 views

Is there a jump condition for this PDE? ( Brinkman model , piecewise constant permeability)

The Brinkman equations for steady flow of an incompressible fluid through rigid porous solid are: $-\dfrac{\mu_0}{k}\mathbf{v} - \mathrm{grad}p + \mu_0 \Delta \mathbf{v} =0$ and ...
1
vote
1answer
101 views

How to solve coupled steady laminar diffusion flame jet problem? [closed]

I am trying to solve governing equations of laminar diffusion flame jet for steady state case. In the next step, I will solve for unsteady case. I have non-dimensional continuity, axial momentum, ...
2
votes
2answers
322 views

Visualising Maxwell's equations using MATLAB

I asked this question to help me understand what is going on in one of Maxwell's equations. I am happy with following through the maths on paper now, but would like to use MATLAB to take it one step ...
0
votes
0answers
140 views

Finite differences and Neumann boundary conditions

I am dealing with a highly nonlinear system of two PDEs. I already have a code to solve the system in case of Dirichlet boundary conditions. The explicit system is: $$ \begin{eqnarray*} \partial_{t}u ...
6
votes
2answers
237 views

Understanding the cost of adjoint method for pde-constrained optimization

I'm trying to understand how the adjoint-based optimization method works for a PDE constrained optimization. Particularly, I'm trying to understand why the adjoint method is more efficient for ...
1
vote
2answers
133 views

use Matlab's PDE toolbox to solve PDE with variable coefficients [closed]

I'm new to the PDE toolbox in Matlab. From the PDE specification window of the toolbox, it looks like one can only solve PDE with constant coefficients. How can ...
2
votes
3answers
211 views

Books on mathematical foundation of finite element methods

After reading three books about finite element method, with two of them covering also finite volume and grid generation, I found myself lost when I have to discuss these topics with library developers ...
3
votes
1answer
144 views

Finite Elements Weak Formulation generalization

I am struggling with an equation that represents the Weak form of Galerkin method: $ \phi^{T}F(\textbf{u})\sim \int_{\Omega}^{ } \phi.f_{0}(\mathit{u},\nabla \mathit{u}) + ...
7
votes
1answer
130 views

Finite element convergence rates for mixed problems

I've coded up a Stokes Flow problem using finite elements and am in the process of verifying that it works. I'm just not sure what convergence rate I should be expecting as I globally refine the mesh. ...
4
votes
3answers
154 views

vector PDEs on manifolds

What are the subtleties involved in solving vector PDEs on manifolds? Can someone suggest a reference summarizing the problems involved? Specifically I want to solve a vector Helmholtz equation with ...
1
vote
2answers
179 views

wavelet for numerical partial differential equations

Is there a good introduction into wavelet Galerkin schemes for numerical partial (and ordinary) differential equations?
1
vote
2answers
139 views

How to compute $\Delta u$ on the boundary of the biharmonic equation?

Let $u$ be the answer of a PDE.Is there any relationship between $u,\frac{\partial u} {\partial n}$ and $\Delta u$. I have the values of $u$ and $\frac{\partial u} {\partial n}$ on $\partial \Omega$ ...
3
votes
1answer
109 views

Time Integration of a nonlinear reaction-diffusion system

I want to solve the following system of nonlinear reaction-diffusion equations (Schnakenberg Turing) using FEM methods (such as deal.ii): $$ \partial_{t} u = \Delta u + \gamma\left(a-u+u²v\right)$$ ...
0
votes
0answers
29 views

Second order PDE with minimum condition

I need to solve 2nd order PDE of $F=F(X,Y)$, where X and Y are functions of t. I have got to point where I transformed my PDE to canonical with logarithmical transformation for X and Y to get constant ...
2
votes
0answers
37 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
votes
1answer
68 views

Guidelines for choosing manufactured solutions for numerical PDE schemes

When testing a numerical method for a PDE, I know that it's often useful to compare it to a known analytical solution. If none is available, one can always 'manufacture' a solution, substitute it ...
9
votes
1answer
472 views

What are the relative benefits of using Adams-Moulton over Adams-Bashforth algorithm?

I am solving a system of two coupled PDE's in two spatial dimensions and in time computationally. Since the function evaluations are expensive, I would like to use a multistep method (initialised ...
3
votes
2answers
140 views

biharmonic equation

I want to solve the biharmonic equation numerically, that is: $$\Delta^2 u=f~~in~~\Omega$$ $$u=g_1~~on ~~\partial \Omega$$ $$\frac {\partial u}{\partial n}=g_2~~on ~~\partial \Omega$$ Using Green's ...
1
vote
2answers
62 views

Performance metrics to compare initial-boundary value problem solutions

I am comparing the performance several finite difference methods of solving an initial-boundary value problem. There are several dimensions to this comparison: Number of cells Number of timesteps ...
2
votes
0answers
108 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 ...
4
votes
1answer
264 views

Coupled nonlinear PDEs with time dependence on the RHS

I would like to numerically solve the following system of 2 coupled partial differential equations for the unknown functions $\psi_X(x,y,t)$ and $\psi_C(x,y,t)$: $\partial_t \psi_X = -i\psi_C - ...
6
votes
1answer
155 views

Solving the quadratic in the Fast Marching Method

The Fast Marching Method is a way of solving the Eikonal Equation on a discrete grid, essentially just computing a wavefront speading out from initial points, IE: The idea is that we want to ...
4
votes
1answer
155 views

Solving a system of nonlinear PDEs by minimization

I have two coupled nonlinear partial differential equations of the form: $ \begin{align} \dot{u} -f(u,u',u'',v,v',v'')=0 \\ \dot{v} -g(u,u',u'',v,v',v'')=0 \end{align} $ The boundary conditions are ...
3
votes
0answers
159 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 ...