Tagged Questions

A partial differential equation that describes diffusion phenomena.

learn more… | top users | synonyms

0
votes
0answers
37 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
44 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 ...
4
votes
2answers
165 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
42 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 ...
0
votes
2answers
104 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 ...
3
votes
1answer
93 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 + ...
1
vote
1answer
92 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, ...
12
votes
3answers
190 views

What is the current state of the art in solving higher dimensional parabolic PDEs (multi-electron Schrödinger equation)

What is the current state of the art for solving higher dimensional (3-10) parabolic PDEs in the complex domain with simple poles (of the form $ \frac{1}{|\vec{r}_1 - \vec{r}_2|}$) and absorbing ...
2
votes
1answer
75 views

Method of lines for inhomogeneous Dirichlet conditions

I understand how to set up the boundary conditions for a steady state problem discretized by Galerkin method, for a time dependent PDE below, $$\frac{\partial}{\partial t} u = c\nabla^2 u + a\nabla u ...
5
votes
1answer
1k 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 ...
6
votes
2answers
213 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 ...
3
votes
1answer
705 views

Numerical solution of non-linear diffusion equation via finite-difference with the Crank-Nicolson method

I want to numerically solve the non-linear diffusion equation: $$ \frac{\partial}{\partial t} T(x,t)= \frac{\partial}{\partial x}\left(T^{5/2} \frac{\partial T}{\partial x} \right) $$ I want to use ...
5
votes
1answer
138 views

Stabilization of solution to one-dimensional system of PDE

I am trying to solve numerically next PDE system: $$\frac{\partial c}{\partial t}=\epsilon\frac{\partial}{\partial x}(\frac{\partial c}{\partial x}+\rho\frac{\partial \varphi}{\partial ...
8
votes
1answer
145 views

How to recreate this result (from a book)?

The result I'm interested in is found within "Synchronization: A Universal Concept in Nonlinear Sciences" page $333$ figure $14.3$. The peculiar fragment is also provided at the end of this post. So ...
4
votes
1answer
100 views

Boundary value technique for heat equation

My heat equation is $$ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}, \quad x \in [0,1], \quad t \in (0,0.1] $$ with initial condition $u(x,0)=\sin(\pi x)$ and homogeneous ...
6
votes
1answer
576 views

Example of a PDE model with nonlinear Dirichlet boundary conditions

Is there any application for PDEs with nonlinear Dirichlet boundary conditions? That is, I am looking for an example of a partial differential equation for a state $u$ posed on a domain $\Omega$ with ...
4
votes
2answers
350 views

First order finite volume spatial discretization of the heat equation on an unstructured triangle mesh

Consider a scalar field $u$ on an unstructured triangle mesh which is constant on each face. Let $A_i$ be the area of triangle $T_i$, $N(i)$ the set of triangles sharing an edge with $T_i$, and ...
4
votes
1answer
141 views

Asymptotic convergence of the solution to a parabolic pde to the solution of an elliptic pde

Suppose I have the parabolic system $$u_t=\nabla\cdot(k(x)\nabla u)+f,\quad (x,t)\in\Omega\times I$$ with dirichlet boundary conditions $$u=g, \quad x\in\partial\Omega$$ and initial condition ...
2
votes
1answer
689 views

Meaning of CFL condition on parabolic problems

I've been studying this FEM theory and for the parabolic problems, there's the analysis of stability of the $\theta$-method. I followed the analysis and they get this CFL (Courant-Friedrich-Lewy) ...
4
votes
1answer
331 views

The effect of the boundary condition on the convergence of the fdm scheme

I know the boundary condition is usually a tricky question. However, I am testing a finite-difference scheme for the equation of the form $$u_t=a(x)u_{xx}$$ that I know the analytical solution of. So ...
4
votes
2answers
138 views

choice of the norm for Crank Nicolson stability estimate

I have a variable coefficient pde of the form $$u_t=c(t,x)u_{xx}, t\in [0,T], x\in [0,1]$$ with initial data $u_0=u(0,x)$ and $c(x,t)\in C([0,T]\times[0,1])$. I use three point discretization for the ...
3
votes
1answer
118 views

intuition behind the different discrete norms for Crank Nicolson

I am solving a heat equation $u_t=Au$ with Crank-Nicolson finite-difference method and $A$ is a usual discretization matrix for $u_{xx}$ term. I want to tell something about the whole error vector ...
2
votes
1answer
116 views

Is it necessary to project the initial condition onto the variational space in a fully discrete galerkin method?

I'm solving a simple 1D heat diffusion problem $$u_t=u_{xx},\quad \Omega\times[0,T]$$ $$u=0\quad, \partial\Omega\times [0,T]$$ $$u(x,0)=f$$using a fully discrete galerkin finite element method. This ...
1
vote
2answers
100 views

how to approach time zero when the equation is not defined at that point

Some background: define a process $Y_t=\frac{1}{t}\int_0^tX_sds$, where $X_t$ is a standard Brownian motion. Then I define a function $u(t,X_t,Y_t)$ and require it to be a martingale. Thus, by ...
3
votes
1answer
68 views

semiboundedness of the operator and it is affect on stability

I remember seeing in the book by Kreiss "Time-dependent partial differential equations and their numerical solution" that if some elliptic differential operator satisfies $$(Lu,u)\leq K(u.u)$$ for the ...
5
votes
1answer
465 views

Is the heat equation with Neumann boundary conditions well-posed?

For example I consider a heat equation that I want to solve numerically : $$u_t=u_{xx},$$ In order to have a uniqueness on a computational bounded domain I have to have boundary condition specified ...
1
vote
2answers
166 views

Smoothing the diffusion coefficient to improve convergence

I have been reading a book by Thomee and he considers the case of $u_t=(au_x)_x$, for the case of $a$ possibly being discontinuous. Then he says that the problems with convergence might occur, and ...
5
votes
3answers
313 views

analyze stability on a nonuniform grid

Assume you have a stability constraint between the space distance in time and space, for example, with an explicit Euler method for $u_t=u_{xx}$ we know $\tau\leq h^2/2$. That is, one can do stability ...
5
votes
1answer
236 views

eigenvalue analysis vs fourier analysis for stability and their equivalence

I have a question regarding stability analysis for constant coefficient pde. Suppose I am looking at the pde $$u_t= au_{xx}$$ So the first approach is to compute the amplification factor, and this is ...
4
votes
1answer
47 views

regularity of a solution and its affect on the global error

I am solving the following equation $u_t=x^2u_{xx}+\frac{x-y}{T-t}u_y$ with an initial data $u_0=max(y-C,0)$ for some $C$ in the domain of the numerical method. In time I am solving that on $[0,T]$. ...
9
votes
1answer
508 views

Optimal use of Strang splitting (for reaction diffusion equation)

I made a strange observation while computing the solution to a simple 1D reaction diffusion equation: $\frac{\partial}{\partial t}a=\frac{\partial^2}{\partial x^2}a-ab$ ...
5
votes
1answer
213 views

Does ADI/Split-operator change the stability properties of the Crank-Nicholson method?

I'm using the Crank-Nicholson method to solve the time-dependent Schrödinger equation with the split-operator method. I'm getting some weird results that are probably the result of a bug somewhere in ...
8
votes
2answers
225 views

Where can I find a good reference for the stability properties of several methods of solving parabolic PDEs?

Right now I have a code that uses the Crank-Nicholson algorithm, but I think that I would like to move to a higher-order algorithm for timestepping. I know that the Crank-Nicholson algorithm is stable ...