Questions tagged [parabolic-pde]
A partial differential equation that describes diffusion phenomena.
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Confusion between standard finite element and mass-lumping finite element methods
Consider the following equation, subject to homogeneous Neumann boundary condition.
$$
u_t = \Delta u + f(u).
$$
The weak formulation is as follows:
$$
(u_t,w) = (\Delta u, w) + (f(u),w), \quad \...
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votes
1
answer
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views
Time discretisation after splitting a 4th order equation
Suppose we have a fourth-order parabolic PDE
$$\partial_t u + a(t)\Delta( \Delta u) - div( b(x,t)\nabla u) =0$$
We split the equation into two second-order equations by introducing $w = \Delta u$ thus
...
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3
votes
0
answers
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views
Form of nonlinear diffusion equation
Consider the following nonlinear diffusion problem,
$$
\frac{\partial u}{\partial t} = x^{-2}\frac{\partial}{\partial x}\left(x^2 u^4 \frac{\partial u}{\partial x}\right), \quad 0 < x < 1
$$
We ...
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1
answer
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How do I use pdepe for a first order parabolic PDE with only one boundary condition?
I am trying to use Matlab's pdepe.m to solve the first order parabolic PDE
$$\frac{\partial u}{\partial x}+\frac{\partial u}{\partial x}=x$$
I have not had trouble coding the argument of pdepe @pdefun:...
6
votes
2
answers
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views
Continuous vs discontinuous space-time FEM
What are some reasons for approximating a (e.g. parabolic) PDE using the space-time method, with continuous finite elements in time, vs discontinuous finite elements in time?
Are there e.g. ...
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2
votes
1
answer
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views
Objective function for PDE-constrained boundary control problem in cylindrical coordinates
I'm interested in solving a boundary control problem for an axisymmetric diffusion problem where diffusive fluxes only appear radially. The corresponding problem for a uni-dimensional slab can be ...
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votes
1
answer
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views
Is the Alternating-Directions Implicit method dependent on the space increment?
I am writing an Alternating-Directions Implicit Method for simple 2D diffusion ( \begin{equation*}
\frac{df(x,y,t)}{dt}=D\Delta u
\end{equation*}). Tridiagonal matrices are solved via Thomas ...
2
votes
0
answers
83
views
Numerical scheme for the heat equation on the icosahedral hexagonal grid
I have a predefined grid(like this) that is spawned from a regular icosahedradron. It consists of many hexagons and 12 pentagons (corresponding to icosahedradron vertexes). I can tweak the granularity ...
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3
votes
1
answer
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Nondimensionalization of a multi-component chemical diffusion equation
Edit I've modified the equations because they were wrong and I added the whole system, as asked by @Wolfgang
I am trying to nondimensionalize a system of partial differential equations similar to 2nd ...
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1
answer
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views
Is this a diffusion equation, or something else？
This is a time fractional PDE，where $0<\alpha<1$.
I would like to know how to describe this equation and where does this equation come from.
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1
answer
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"Optimal" domain partitioning in domain decomposition algorithms
When solving a PDE numerically by domain decomposition methods, what is the "optimal way" to split the domain? Are there any results stating that a particular partition of the domain yields &...
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votes
1
answer
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Derivation of a parabolic PDE using Alternating Direction Implicit method
I have a very simple question concerning Alternating Direction Implicit (ADI) Scheme.
If I have an equation of the form:
\begin{equation*}
\frac{df(x,y,t)}{dt} = \nabla^2 f(x,y,t) + f(x,y,t)
\end{...
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1
vote
1
answer
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views
3d schrodinger equation weak form
SCHRODINGER’S EQUATION
$$-ih u_{t}(x,y,z,t) = \frac{h^2}{2m} u_{xx}(x,y,z,t)+ \frac{e^2}{r}u(x,y,z,t)$$
The potential $\frac{e^2}{r}$ is a variable coefficient.
So, let’s take the free Schrodinger ...
user38211
0
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0
answers
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The physical meaning of conservative mass in diffusion equation
I am working on 1-D mass transient diffusion in a radial domain (spherical object) using finite volume method. My equation reads
$$\frac{\partial C}{\partial t} = D\left[\frac{\partial^2 C}{\partial r^...
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4
votes
2
answers
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views
Is there a simple way to add a sparse matrix to an LU decomposition of a dense matrix?
I am solving a parabolic equation in the form:
$$
\left( {M \over\tau_j} + A \right) u^{j+1} = M f^j + {u^j \over \tau_j},
$$
where $A$ and $f$ are a dense stiffness matrix and the right hand side of ...
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votes
2
answers
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views
Two-dimensional heat equation with Neumann boundary conditions: any hope to find an analytical solution?
I am looking for references showing how to analytically solve the heat equation with Neumann boundary conditions in two dimensions.
So far, I have found the problem solved analytically in one ...
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votes
1
answer
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views
Verification of order of convergence of Implicit Euler Method to numerically solve Black-Scholes PDE
I'm trying to verify the order of convergence for implicit Euler method to numerically solve Black-Scholes PDE. Theory says that it should be $O(\Delta t + \Delta S^2).$ My code is working absolutely ...
user36184
4
votes
1
answer
790
views
Backwards Difference Implicit Method for Nonlinear Parabolic PDE in Python
Original Stack Overflow Question: https://stackoverflow.com/questions/65683788/indexerror-index-31-is-out-of-bounds-for-axis-1-with-size-31?noredirect=1#comment116218335_65683788
PDE: u_t = u_xx + u(...
0
votes
1
answer
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How to insert a(x) function in non homogeneous parabolic pde for implicit method in Python?
I have the following inhomogeneous parabolic initial/boundary value problem:
$$u_{t}(t,x) = (1-x^{2})u_{xx}(t,x)+u(t,x),$$ for $t \in [0,1]$ and $x \in [-1,1]$
$$u(0,x) = \sin(\pi x),$$ for $x \in [-...
user37062
1
vote
1
answer
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views
Non-Linear advection diffusion with nondifferetiable advection term
I'm looking at Murray's book: Mathematical biology: an introduction , first volume, pag. 404
In particular, I'm interested to solve the following PDE:
$$\partial_t u = \partial_x (\text{sign}(x) u) + \...
- 309
4
votes
2
answers
418
views
1D FEM for nonlinear diffusion coefficient
I want to solve with linear finite elements the equation $$\partial_t u = \partial_{x}(a(u)\partial_xu)$$
in the domain $t \in [0,1]$ and $x \in [-L,L]$. Here $a(u)$ is just a function of $u$.
...
- 309
1
vote
0
answers
50
views
Effect of reducing flux consistency order at boundary on convergence order
Consider the 1D nonstationary convection-diffusion PDE
$$
\begin{alignat}{2}
\partial_t u &= -a \partial_x u + D \partial_{xx}u, &\qquad x \in (0,1), t \in (0,T), \\
f(t) &= \left.\left( a ...
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votes
0
answers
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views
fixed point iteration on DD method
I have to solve the the problem $u_t+\Delta^2u=f(u)$, where $f(u)$ is non-linear, using domain-decomposition method.
My approach is first using fixed point iteration on mixed form i.e to say $u^{k+1}...
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votes
0
answers
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views
Convergent Finite Difference Scheme for Parabolic Equation
Consider the PDE $$u_t = b_{11}u_{xx} + 2b_{12}u_{xy} + b_{22}u_{yy},$$
where $b_{11}, b_{22} > 0$, and $b_{12}^2 < b_1b_2$.
In Strikwerda's book, the ADI scehme \begin{align*}
\left( 1 - \frac{...
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1
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views
Numerically solving nonlinear parabolic stochastic PDEs
For a project I'm doing, I have to numerically solve a nonlinear parabolic stochastic partial differential equation, of the form
$$
u_t = u_{xx} + f(u)(u_x)^2 + a(u) + b(u)W(t, x),
$$
where primes ...
user34137
1
vote
1
answer
89
views
Stability condition for explicit time FEM for parabolic pdes
If we discretize a parabolic pde to obtain the system of ODE's
$\frac{\boldsymbol{B}}{\Delta t} \boldsymbol{u}_k = (\boldsymbol{K} + \frac{\boldsymbol{B}}{\Delta t}) \boldsymbol{u}_{k-1} + \boldsymbol{...
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votes
1
answer
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views
Type of Rosenbrock method by its coefficients
A Fortran code that solves stiff PDE systems contains the following arrays of Rosenbrock-Wanner method coefficients:
...
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vote
1
answer
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views
How suitable is multigrid method for time-dependent PDEs?
For elliptic PDEs (Poisson-type), the multigrid method is very sufficient, but how about time-dependent problems (i.e parabolic or hyperbolic PDEs)?
Is it efficient to solve such problems using a ...
- 481
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1
answer
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Imposing periodic boundary condition for linear advection equation - Node problem
I've spent the whole day trying to figure out what is the correct way to impose (and implement) periodic boundary conditions $u(0,t)=u(1,t)$ for all $t>0$ for the simple advection equation $u_t + v ...
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0
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views
Solving Parabolic PDE using Matlab
I have the following pde (Burger's equation)
for $\epsilon>0: u_t+u.u_x=\epsilon.u_{xx}$ and $x\in \mathbb{R},t>0$
and the initial condition: $u(x,0)=\phi(x)=1_{(-\infty,0)}(x)$.
I want ...
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1
answer
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views
Dirichlet boundary conditions in the 1D Heat Equation
Please consider the assignment I have uploaded on the picture.
I am confused about the functions $g_L(t)$, $g_R(t)$ and $\eta(x)$. What are they and how do I find them... My question:
Is it possbile ...
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votes
1
answer
406
views
Numerically solving a non-linear PDE
I have this non-linear partial differential equation.
$$
\frac{\partial C}{\partial t}=\left(\frac{\partial C}{\partial x}\right)^2+C\frac{\partial^2 C}{\partial x^2}
$$
I want to use the finite ...
- 33
1
vote
1
answer
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views
Error for the finite differences scheme -- Advection equation
Consider the advection equation (1D in space)
$$
\frac{\partial u}{\partial t} + V\, \frac{\partial u}{\partial x}=0
$$
and we solve it numerically on $[0,1]\times [0,1]\ni (t,x)$ using a forward ...
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vote
0
answers
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views
Oscillations when solving parabolic heat equation with FTCS
I'm wondering if someone could help me out, or point me in a direction of how I can understand the following oscillations that occur when I solve the Porous Medium Equation $$u_t = u_{xx}^{m+1}$$ ...
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votes
0
answers
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views
Analytic vs discrete understanding of PDE
The PDE I am working with:
$$\partial_tu = \nabla \cdot (a(x)\nabla u)-\beta(x)u\\
\partial_nu=0, x \in \Omega \subset \mathbb{R}^2\\
\beta(x)>0$$
Integrate the PDE:
$$\int_\Omega \partial_t u=\...
- 216
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votes
0
answers
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views
What will PDE discretization matrix look like for time and space? [duplicate]
Please note: this question is not a duplicate of this question since, while the PDE is the same, the nature of this question is different, i.e. the other question treats a different aspect of this PDE....
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0
answers
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views
Discretizing a parabolic PDE with finite volume method
I want to discretize the following parabolic PDE:
$$u_t = \nabla\cdot(\alpha(x)\nabla u)- \beta u\\
x\in\Omega \subset \mathbb{R}^2\\
\partial_n u = 0\\
u(t,0) = u_0(x)\ge 0, \alpha(x)>0$$
Given ...
- 216
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votes
0
answers
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views
Unusual boundary conditions on Matlab
I'm trying to solve the following PDE by Matlab,
$$
u_t-\Delta u = 0, \quad \text{in}\quad \Omega\times (0,T)
\tag{1}
$$
$$
u_t-\Delta_\Gamma u + \partial_\nu u=0,\quad \text{on}\quad \Gamma\times(0,...
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vote
1
answer
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views
Physical data for heat equation
I want to implement an algorithm to solve a heat equation, i.e.
\begin{align*}
\partial_t u - \Delta u = f \text{ in } \Omega\times(0,T)\\
\partial_nu = 0 \text{ in } \partial\Omega \times (0,T)\\
u(0)...
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votes
1
answer
309
views
PDEPE nonlinear
I would like to use Matlab's pdepe to solve this system:
$$ s_t =(sr)_x + s_{ xx } \\
r_t =(\frac{ A }{ B }r^2+s)_x + \frac{ A }{ -K } r_{ xx } $$
where $A$, $B$ ...
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0
answers
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views
Accuracy of finite difference method for heat equation on a disk
To study an approximation for the heat equation $$\frac{\partial^2 u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2}\frac{\partial^2 u}{\partial\theta^2}=f(r,\theta)$$
on the ...
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votes
1
answer
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views
Parabolic differential equations with time delay
Let $d_1=1,d_2=2,a_{11}=\frac{5}{13},a_{12}=\frac{22}3,a_{21}=-2,a_{22}=\frac{6}7,\tau=\frac{5}7$, $\psi(t,x)=\cos^42x,\phi(t,x)=\frac{3}{13}x^4\sin^2 3x$, $\Omega=[0,200]$
How to solve:
$$\left\{
\...
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2
answers
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views
Trying to compute the error from comparing two arrays
Some context: I am working with the Black-Scholes model.. I have an explicit (Black-Scholes) formula which is the exact solution to my problem. I have written code which implements a finite-difference ...
- 103
1
vote
0
answers
73
views
Growing error from a smooth initial condition for Fisher KPP equation
I'm studying the Fisker-KPP equation on the line (and in $]0, 100[$ numerically):
$$
\partial_t u = \Delta_{xx} u + u(1-u)
$$
I notice a behavior I don't understand with a smooth initial condition $...
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votes
0
answers
133
views
How one could choose the value of viscous coefficient for obtaining stable solution of Burgers' equation?
Burgers' equation is a fundamental PDE used in various fields such as number theory, gas dynamics, heat conduction, elasticity, etc. It is crucial especially for developing numerical models for ...
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vote
0
answers
93
views
Implementing initial conditions into the solution domain of a 1-D advection-diffusion equation
I have the following PDE.
$\frac{\partial c}{\partial t} + v\frac{\partial c}{\partial x} - D\frac{\partial^2 c}{\partial x^2} = 0
\\$.
I have discretized it such that i now have
$\frac{dC}{dt} = ...
- 11
2
votes
1
answer
199
views
An interesting numerical pde problem
I'm somewhat struggling with how they are getting this scheme. This is a problem from Morton & Mayers book on numerical pde solutions. I think they are using a forward difference approximation for ...
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votes
1
answer
273
views
Cauchy problem with a change of variables
Consider an advection-diffusion problem
$$u_t+au_{xx}+bu_{x}=0,\quad x>0.$$
Now I want to remove the drift and rewrite the problem for the new domain. I use the change of variables $y=x-bt$, and ...
- 1,206
0
votes
1
answer
681
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spurious oscillations Crank-Nicolson
I want to make sure I am clear with all the reasons why oscillations and how wild they get. I will summarize what I understood from the books with respect to the application of the Crank-Nicolson to ...
- 1,206
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vote
4
answers
448
views
discretization error for the explicit scheme for heat equation
I am a little confused about the truncation error and what it actually means. So, For example I have an explicit approximation to the heat equation
$$\frac{u_{j}^{i+1}-u_{j}^{i}}{\delta t}=\frac{u_{j-...
- 1,206