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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2
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0answers
68 views

Segregated solving of a tightly coupled system of PDEs

To compute the evolution of a free surface between two incompressible, immiscible liquids, two tightly coupled equations have to be solved, the volume fraction advection and the Navier-Stokes ...
5
votes
0answers
88 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 ...
0
votes
1answer
126 views

Parallel 2d heat equation (implicit timestepping) using MPI

I am trying to solve the time dependent heat equation with backward euler timestepping and second order space finite differences. This results in a Poisson system needing to be inverted. In serial ...
2
votes
1answer
91 views

Von Neumann stability analysis in 3d

I need to get a stability criterion for the numerical scheme for equation $$\frac{\partial u}{\partial t}-\frac{\partial^2 u}{\partial x^2}-\frac{\partial^2 u}{\partial y^2}-\frac{\partial^2 ...
2
votes
1answer
100 views

Courant Friedrichs Lewy condition - how to get it?

I am interested, how can we get CFL condition for every type of PDE? It's known that for 1st order linear equation $$\frac{\partial u}{\partial t}+a\frac{\partial u}{\partial x}=0 $$ CFL is get from ...
1
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0answers
115 views

Derivation of a Higher Order Compact Alternating Direction Implicit Method

I dont understand how this Higher Order Compact ADI scheme, which is fourth order in time and space, for the wave equation is derived: I go through the following Using Tylor's expansion $u(t+h,x,y)$ ...
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0answers
52 views

Locally conservative method for differential generalized eigenvalue problem

I have to approximate the smallest eigenvalue of the following generalized eigenvalue problem $$ - \nabla \cdot D(x) \nabla p(x) + \alpha(x) p(x) = \lambda \beta(x) p(x) $$ over a domain like ...
4
votes
2answers
92 views

Discretization method for a reaction dominated elliptic PDE

I'm working with an elliptic reaction diffusion PDE of the form $$-k\nabla^2u+cu=0$$ I've noticed that when the reaction term dominates over the diffusion (i.e. $c>>k$), the true (exact) ...
7
votes
2answers
139 views

The effect of decoupling a coupled system of PDEs

I asked a somewhat similar question previously but perhaps it might have been too specific for anyone to really answer. Here is a bit more general of a question that I am struggling with. Consider the ...
4
votes
1answer
130 views

User Friendly GUI based 3d PDE solving software

I'm searching for software which can solve 3d IBVP like PDEtool in Matlab (user-friendly GUI, easy to learn how to use). What could you advise to me? (ANSYS,COMSOL,OpenFOAM are too difficult for me)
6
votes
2answers
144 views

Spectral Methods in time

I was reading up on Spectral Methods for PDEs. In all the descriptions I read, while the position component is approximated via a Fourier series or other methods, the time component is still ...
5
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0answers
93 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: $$ ...
2
votes
1answer
107 views

One finite difference scheme

There is PDE: $$\frac{\partial u(r,\varphi,\psi,t)}{\partial t}=\operatorname{div}A(r,\varphi,\psi)\nabla u +f(r,\varphi,\psi,t) $$ We solve numerically IBVP for the ball $B_{1}(0)\subset ...
1
vote
0answers
51 views

Stability question (finite difference): dealing with corner nodes

Consider one initial boundary value problem for sphere. $$\frac{\partial u}{\partial t}=\operatorname{div}A\nabla u +f$$ Here is explicit numerical scheme (we consider that it is stable): ...
2
votes
1answer
156 views

Implementation of gradient zero boundary conditon in advection-diffusion equation

My question is about Finite Element Method. I want to know how to implement "gradient zero" conditions to advection-diffusion equations in conservative form like, $\frac{\partial \rho}{\partial t} + ...
1
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0answers
41 views

how to find outward normal for robin codition [duplicate]

I wrote the code to fix this problem; now I should validate it using a test function to see the error in my code. I can't figure out how starting from u can derive the function g on the edge, in ...
-1
votes
1answer
124 views

Pde problem with robin boundary condition

I have my pde 2D problem with robin condition (form: du/dn +ku=g) to solve with matlab. i have the exact function u and I want to find the function g in robin condition. How can i do it? thanks for ...
2
votes
1answer
81 views

variational formulation

I would like to minimize: $$J = \int_{\Omega} \|\nabla u - \nabla g\|^2 + \lambda \|\frac{\partial u}{\partial t} + \nabla u.v||^2 ~\text{dx dy dt}$$ where $u(x,y,t)$ is the unknown function, the ...
0
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0answers
74 views

Calculating Divergence in COMSOL

Is it computationally safe and accurate to use the following equation in COMSOL to compute the divergence of the vector quantity J (instead of using its general built-in equations that have $\nabla$ ...
9
votes
3answers
308 views

How to deal with curved boundary condition when using finite difference method?

I'm trying to learn about numerically solving PDE by myself. I've been beginning with finite difference method(FDM) for some time because I heard that FDM is the fundament of numerous numerical ...
0
votes
1answer
132 views

What libraries provide an implementation of multigrid?

I am working on numerical method of Multigrid. What's the available implementation(solver) (actually used in scientific computation) of multigrid method?
4
votes
3answers
148 views

Is there a general framework for solving PDEs on uniform grid in parallel

Hej, I want to simulate a partial differential equation (a modified Cahn-Hilliard equation, but the details do not matter much. The questions also applies to the diffusion equation). I'm looking for ...
0
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0answers
197 views

Magnetostatic and magnetodynamic problems in freefem++

I would have liked to use freefem++ for implementing some simulations as in the title but I didn't find anything about it. I know it does implement Nedelec elements but I don't know for example how to ...
3
votes
3answers
456 views

Stuck on a hideous differential equation

I'm stuck on this PDE: $$\rho \cdot C \cdot \dfrac{\partial T}{\partial \tau} + u \cdot \rho \cdot C \cdot \dfrac{\partial T}{\partial r} = \dfrac{\lambda}{r^{2}} \cdot \dfrac{\partial}{\partial ...
1
vote
3answers
248 views

Capacitance in freefem++

I would like to simulate a capacitor in 2d with freefem++. This is the code I used: ...
5
votes
1answer
720 views

How to add reaction and source terms to a diffusion PDE solver written with MATLAB's pdepe function?

I have the following system of equations which I'm trying to solve using Matlab's pdepe solver. The 1-D spherical heat diffusion equation with heat generation ...
0
votes
0answers
55 views

Using Lattice Boltzmann for something different from Navier Stokes

First of all: is it possible to use the Lattice Boltzmann method if no equilibrium distribution exists? Secondly, if I have a PDE, how do I derive the corresponding collision function?
3
votes
1answer
321 views

Stability of numerical method for 1D Burger's equation

I am trying to solve 1D viscous Burger's equation numerically and I cannot apply von Neumann analysis because the equation is non-linear. How do I predict the stability criteria for my system? I also ...
6
votes
2answers
161 views

Is there a good tutorial or textbook-like source on implementing ENO/WENO with limiters in one (and more than one) dimension?

I've inherited a finite volume code that does a second-order discretization of flux terms for a set of mixed parabolic-elliptic equations with discontinuous diffusion coefficients. The impression I ...
5
votes
0answers
139 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: ...
0
votes
1answer
105 views

Can the method of lines technique be used to solve a ODE directly for the steady-state value?

Say we have a discretised a coupled nonlinear system of two PDEs to give a system of ODEs which approximates the original system, $$ \frac{\partial u}{\partial t} = F_1(t,\boldsymbol{u,v}) \\ ...
8
votes
3answers
223 views

Are there finite element software who handles more than five dimensions?

I'm a beginner with FE. My application is the pricing of financial derivatives where the space is five dimensional. So, adding time, the problem has six dimensions. I tried to look around (Fenics, ...
6
votes
1answer
438 views

Automatically generating finite difference matrices for systems of PDEs

Suppose that you have a system of PDEs to solve. At least for simplicity, let's assume it's time independent, quasi-linear (linear in its derivatives) solved on a rectangular grid in (x,y) space, and ...
1
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0answers
24 views

Local matrices with complement function

the title is a bit obscure and I should clarify. In an other post I asked for the formulation of local matrices coefficients and it turned out that to compute the local mass matrix we have : For a ...
3
votes
3answers
124 views

Smoothly varying dense matrices arising from computational science

I have written an algorithm to solve a dense system with smoothly varying entries. This means I assume there is no large jump from any entry to its neighbors. I would love to use ...
2
votes
2answers
291 views

How to solve an advection-diffusion equation

I need to solve an advection-diffusion equation of the form: $\frac{∂u}{∂t}=\frac{1}{x}\frac{∂u}{∂x}+\frac{∂^2 u}{∂x^2 } $ with MATLAB. Could you guide me, please? Is the Crank-Nicolson method ...
1
vote
1answer
159 views

How to write this non-linear PDE with the finite volume method?

I wish to solve a coupled system of non-linear equation of this form, $$ u_t = -(\mathcal{F})_x + f(x,u,w) \\ w_t = \mathcal{F} + g(x,u,w) $$ by stepping the equations forward in time. The first ...
8
votes
1answer
138 views

Can I solve this time-independent PDE by adding a time derivative and marching in time?

I want to solve this PDE: Currently I have some code that will automatically generate pde solutions for a very similar pde that includes a time derivative (partial d/ partial t) using an ADI ...
3
votes
1answer
227 views

weak formulation of coupled pdes for fenics

I am trying to implement the following system of time-dependent, coupled nonlinear pdes in FEniCS: ...
2
votes
1answer
264 views

export pde solution to non-uniform grid

After exporting the solution (u) and the mesh (p, e, t) from pdetool to workspace, I need to compute the solution at a set of predefined points. These predefined points where the solution is sought ...
3
votes
4answers
183 views

Request algorithm recommomendation for 2D generalised Poisson solution

I need to examine the (static) electric field distribution around various electrode configurations in the presence of a dielectric, and have been using a finite difference approach to the 2D ...
0
votes
0answers
312 views

Simulink PDE (MATLAB)

I need to build a system in Simulink that solves a PDE, but I can't find any literature or books where it is described how to do it (especially any stuff according to modeling PDE in Simulink). ...
10
votes
3answers
523 views

What is the purpose of using integration by parts in deriving a weak form for FEM discretization?

When going from the strong form of a PDE to the FEM form it seems one should always do this by first stating the variational form. To do this you multiply the strong form by an element in some ...
1
vote
1answer
339 views

time-dependent nonlinear pde in fenics

I am interesting in solving the following nonlinear, time-dependent pde in 2 spatial dimensions (complex Gross-Pitaevskii eq): $$i \frac{\partial \psi}{\partial t} = \left[ -\nabla^2 + (1-i ...
7
votes
1answer
181 views

Navier-Stokes solver: How to adjust the time step based on non-linear terms?

My code solves the incompressible Navier-Stokes equation in a conducting fluid, together with the induction equation: $ \partial_t u + u \nabla u + 2\Omega \times u = -\nabla p + \nu \Delta u + ...
0
votes
0answers
292 views

solving a set of coupled differential equations in matlab

plot h vs t for the below coupled equations: $a_1\frac{d^2p}{dx^2}-b_1\frac{d^2g}{dx^2} = c\frac{dh}{dt}$ $a_2\frac{d^2p}{dx^2}-b_2\frac{d^2g}{dx^2} = c\frac{dh}{dt}+d\cdot g$ where, a1,b1,a2,b2 ...
2
votes
2answers
1k views

Time stepping in comsol multiphysics

I have an important question I wasn't able to find an answer to. i would like to know which is the algorithm comsol uses in order to correct the time step it uses. For example, when you try to solve ...
2
votes
2answers
73 views

Quantification of non-stationarity of PDE solution

Suppose I have a time-dependent PDE discretized by the Rothe method and FEM, like $$ \int_{\Omega} k^{n+1/2}(u^{n+1}-u^{n}) v \;\mathrm{d}x = F^{n+1/2}(u^{n+1},u^n)[v] \quad \forall v\in V_h^n. $$ ...
2
votes
2answers
198 views

Flux across a non-boundary line segment in FEniCS

I am solving an elliptic boundary value problem on a subset of the rectangle [-1,1]x[-1,1]. The domain contains the line segment x=0, however this does not need to be a part of the boundary, so it is ...
4
votes
3answers
206 views

Literature material discussing generic frameworks to solve PDEs

I am trying to gather literature material to study how people propose to implement generic frameworks to solve partial differential equations in C++. Despite my effort to search the web, the only ...