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Questions tagged [parabolic-pde]

A partial differential equation that describes diffusion phenomena.

2
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0answers
47 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,...
1
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0answers
87 views

Finite differences for the one-phase Stefan problem

I am trying to code the one-phase, one-dimensional Stefan problem using finite differences in Matlab, similarly to what has already been done in Mathematica (see https://mathematica.stackexchange.com/...
1
vote
1answer
64 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)...
0
votes
1answer
134 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 $A$, $B$ ...
1
vote
0answers
138 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 ...
0
votes
2answers
85 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\{ \...
0
votes
2answers
109 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 ...
1
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0answers
47 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 $...
3
votes
0answers
114 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 ...
1
vote
0answers
78 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} = ...
2
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1answer
193 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 ...
0
votes
1answer
246 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 ...
0
votes
1answer
313 views

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
vote
4answers
209 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-...
0
votes
2answers
451 views

Help implementing finite difference scheme for heat equation

I am trying to solve the following problem via a finite difference approximation: $u_t = k \, u_{xx}$, on $0 < x < L$ and $t > 0$; $u(0,t) = u(L,t) = 0$; $u(x,0) = f(x)$. I take $u(x,0) = ...
1
vote
0answers
33 views

explicit scheme stability restriction

Looking at the plain heat equation $u_t=u_{xx}$ the explicit scheme for it would look like the following iteration: $$u_{m,n+1}=\rho u_{m-1,n}+(1-2\rho)u_{m,n}+\rho u_{m+1,n}$$ I noticed this equation ...
2
votes
1answer
632 views

heat equation on bounded and unbounded domain

I have been reading about the heat equation and I am confused about uniqueness in the case when the domain is bounded and when it is not. In the book I am following, it is common to write the heat ...
4
votes
1answer
123 views

Space-time Galerkin of Burgers changes the convection speed

tldr: Can space-time Galerkin schemes applied to convection-diffusion problems lead to effects on the convection velocity? For time $t\in (0,1)$ and the spatial variable $\xi \in (0,1)$, I am ...
1
vote
0answers
69 views

Stability of two PDEs

I have the following two PDEs which I want to check for stability: $$u_t= u_{xx} , \ u(x,0)=1 , \ u_x(1,t)=-hu^4(1,t) , \ u_x(0,t) = 0 $$ $$u_t = u_{xx}-\sin(x+t)+\cos(x+t) , \ u(x,0)=\cos(x) ,\ u_x(...
2
votes
3answers
382 views

Algorithm suggestion for PDE - example: heat equation

I want to solve the PDE equation numerically. For this, I started my study with something simple; heat equation $$ \frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial^2 x} $$ with the initial ...
0
votes
2answers
258 views

Looking for a matlab/maple code for plotting the truncation error

On page 18 on this text: http://www.dima.uniroma1.it/users/lsa_adn/MATERIALE/FDheat.pdf , the graph in figure 8 on this page, how would I write a suitable code in matlab or maple that will produce ...
1
vote
0answers
116 views

Boundary Conditions for the given PDE

I'm working on the Black-Scholes equation, but I'm pretty new to financial modeling. Right now, I am trying to understand the Black-Scholes PDE. I understand that the Black-Scholes equation is given ...
1
vote
1answer
813 views

Heat Equation - PDE

I'm trying to model the Black-Scholes Equation (transformed into a heat equation) using method of lines in Python. The transformed formula is basically \begin{equation*} \frac{\partial u}{\partial ...
2
votes
0answers
164 views

Modeling First Order Parabolic PDE (Battery Storage Model)

I'm trying to solve the following first order parabolic partial differential equation, \begin{equation*} X \frac{\partial V}{ \partial Q} = -\frac{1}{2} \sigma^2 \frac{\partial^2 V}{\partial X^2} + ...
6
votes
1answer
237 views

Do the class of PDEs that lack initial conditions have a name?

I am trying to think of what this kind of problem is called. I illustrate it with a telegrapher's equation with (hopefully) standard notation. Find $u:\Omega\times \mathbb{R} \to \mathbb{R}$ such ...
1
vote
2answers
109 views

Relation between Time dependent problem and advection diffusion

Is there a relation between say the heat equation $u_t -\Delta u = f.$ and advection-diffusion equation $-\Delta u + c \cdot \nabla u = f$? I have heard several people use this argument in many talks ...
11
votes
2answers
6k views

Periodic boundary condition for the heat equation in ]0,1[

Let us consider a smooth initial condition and the heat equation in one dimension : $$ \partial_t u = \partial_{xx} u$$ in the open interval $]0,1[$, and let us assume that we want to solve it ...
2
votes
0answers
541 views

Solving a system of 4 coupled PDEs representing variable diffusivity

I have four partial differential equations representing mass conservation of two compressible fluid phases (marked by subscripts $p1$ and $p2$) in two different continuum media (marked by subscripts $...
1
vote
1answer
2k views

Implementation of 1D Advection in Python using WENO and ENO schemes [closed]

I'm trying to implement 1D advection solver using WENO and ENO schemes. \begin{equation} \frac{\partial u}{\partial t} + \frac{\partial f(u)}{\partial x} =0 \end{equation} where: \begin{...
0
votes
1answer
114 views

Heat equation with harmonic source and Neumann boundary

Is there an analytic solution or series approach to the 2D time-dependent heat equation in a square with a harmonic source and Neumann boundary conditions $\nabla u.n=0$?
0
votes
1answer
244 views

Adaptive mesh refinement basic conceptual problem

I am a beginner in adaptive mesh refinement (AMR). After I am done with the first two papers by Dr. Marsha Berger, I was trying to write my own code for a problem which has a parabolic partial ...
2
votes
2answers
288 views

FEM for a nonlinear parabolic PDE

I'm looking to numerically compute the solution to $$ k(x,u) \partial_t u - \Delta u = f \quad\quad\text{ in } \Omega \times [0,T]$$ where $k$ is a continuous but nonlinear (in $u$) real-valued ...
1
vote
0answers
354 views

Finite Difference in Polar Co-ordinates

Background Please note that I am duplicating the question on scicomp. I have already asked this in math. I am trying to come up with a scheme in Polar Co-ordinates for the following PDE: PDE I am ...
2
votes
1answer
97 views

Application of CLAWPACK to Richards' equation

I'm looking to solve the Richards' equation. This models water flow in porous media and is a nonlinear, possibly degenerative, parabolic differential equation that takes the form $\partial_t \Theta(\...
2
votes
0answers
130 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 ...
5
votes
2answers
2k 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
102 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 ...
1
vote
3answers
1k views

Applying the method of lines to parabolic PDEs: references and software

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
252 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 + f(x,t)...
1
vote
1answer
250 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, ...
11
votes
3answers
288 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
526 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
4k 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 (...
7
votes
2answers
702 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 ...
4
votes
1answer
2k 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
172 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 x}+\frac{vc}{1-...
8
votes
1answer
177 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
161 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 ...
7
votes
1answer
1k 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 $...
5
votes
2answers
907 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 $L_{...