A partial differential equation that describes diffusion phenomena.
2
votes
1answer
48 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 ...
5
votes
1answer
82 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 ...
3
votes
2answers
91 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
61 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
201 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) ...
5
votes
1answer
126 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
96 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
86 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
99 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
89 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
64 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
295 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
96 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
215 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
131 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
43 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 $[0,T]$. ...
6
votes
0answers
254 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
138 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 ...
7
votes
2answers
165 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 ...
