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Questions tagged [parabolic-pde]

A partial differential equation that describes diffusion phenomena.

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2 votes
1 answer
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Where am I making a mistake in solving the heat equation using the spectral method (Chebyshev's differentiation matrix)?

I would like to numerically solve the following heat equation problem: $$ u_t = \Bigg(2{a \over l}\Bigg)^2 u_{xx} \tag 1$$ $$ x \in [ -1, 1 ] \tag 2$$ $$ u(x, 0) = 0 \tag 3$$ $$ u(1, t) = A \sin \Bigg(...
FriendlyNeighborhoodEngineer's user avatar
3 votes
0 answers
74 views

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 \...
Owen Jun's user avatar
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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 ...
Thede's user avatar
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3 votes
0 answers
154 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 ...
IPribec's user avatar
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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:...
Walden Marshall's user avatar
5 votes
2 answers
325 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. ...
Lilla's user avatar
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2 votes
1 answer
113 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 ...
IPribec's user avatar
  • 617
0 votes
1 answer
191 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 ...
Roman Kirillov's user avatar
2 votes
0 answers
86 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 ...
FrogOfJuly's user avatar
3 votes
1 answer
111 views

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 ...
Iddingsite's user avatar
0 votes
1 answer
93 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.
Shawn D's user avatar
3 votes
1 answer
96 views

"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 &...
user298455's user avatar
2 votes
1 answer
139 views

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{...
Iddingsite's user avatar
1 vote
1 answer
211 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 ...
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0 votes
0 answers
44 views

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^...
Matt's user avatar
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4 votes
2 answers
296 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 ...
Dimitar Slavchev's user avatar
2 votes
2 answers
396 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 ...
Max_89's user avatar
  • 61
0 votes
1 answer
444 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 ...
user avatar
4 votes
1 answer
861 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(...
AlphaArgonian's user avatar
0 votes
1 answer
132 views

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 [-...
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1 vote
1 answer
159 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) + \...
Vefhug's user avatar
  • 309
4 votes
2 answers
463 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 $u$. ...
Vefhug's user avatar
  • 309
1 vote
0 answers
52 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 ...
cos_theta's user avatar
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0 answers
86 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}...
420's user avatar
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2 votes
0 answers
37 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{...
Mike D's user avatar
  • 141
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1 answer
174 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 ...
user avatar
1 vote
1 answer
92 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{...
secondrate's user avatar
3 votes
1 answer
182 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: ...
sequence's user avatar
  • 226
1 vote
1 answer
210 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 ...
spyros's user avatar
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1 vote
1 answer
2k views

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 ...
FEGirl's user avatar
  • 405
0 votes
0 answers
73 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 ...
Robert-ben's user avatar
1 vote
1 answer
866 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 ...
k.dkhk's user avatar
  • 255
0 votes
1 answer
431 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 ...
Peter's user avatar
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1 vote
1 answer
261 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 ...
Smilia's user avatar
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1 vote
0 answers
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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 $$u_t = u_{xx}^{m+1}$$ ...
Gjert's user avatar
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0 votes
0 answers
84 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=\...
sequence's user avatar
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0 answers
38 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....
sequence's user avatar
  • 226
1 vote
0 answers
65 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 ...
sequence's user avatar
  • 226
2 votes
0 answers
81 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,...
S. Maths's user avatar
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1 vote
1 answer
90 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)...
FredBrunck's user avatar
0 votes
1 answer
318 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 $A$, $B$ ...
Nana's user avatar
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1 vote
0 answers
328 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 ...
Daniele's user avatar
  • 11
0 votes
1 answer
119 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\{ \...
LCFactorization's user avatar
0 votes
2 answers
2k 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 ...
Javier's user avatar
  • 103
1 vote
0 answers
74 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 $...
bela83's user avatar
  • 443
2 votes
0 answers
135 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 ...
Loading...'s user avatar
1 vote
0 answers
96 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} = ...
user22328's user avatar
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 ...
bdp's user avatar
  • 21
0 votes
1 answer
279 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 ...
Kamil's user avatar
  • 1,176
0 votes
1 answer
722 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 ...
Kamil's user avatar
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