Partial differential equations (PDEs) are equations that relate the partial derivatives of a function of more than one variable. This tag is intended for questions on modeling phenomena with PDEs, solving PDEs, and other related aspects.

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Matlab PDE toolbox

I'm new at matlab and I can't figure out how to make external temperature not constant but as a function. I wanted to use pde toolbox, I choose parabolic type. My external temperature should be ...
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1answer
49 views

Instability of pdepe in Matlab… boundary conditions?

here is a Matlab beginner banging his head on the wall... I am trying to solve a system of partial differential equations in Matlab, with both derivatives in time and space domains. I am using the ...
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1answer
119 views

Boundary conditions in conforming Galerkin method for biharmonic equation

I am trying to solve simple scalar biharmonic equation using bubnov-galerkin finite element method. I am using $H^2$ conforming basis functions. I was wondering that if anyone can give me some ...
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Temporal convergence of 2D transient advection-diffusion problem

I have a simple 2D transient linear Advection-diffusion problem with zero source term. And I assume a very large diffusion term (which is very stiff) If I use a standard Galerkin discretization in ...
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1answer
67 views

Solving a PDE using Matlab (with varying initial conditions)

I want to solve a 1-D heat conduction PDE using Matlab which looks like $$ \rho c_p \dfrac{\partial T}{\partial t} = \dfrac{\partial}{\partial z}\left( \lambda \dfrac{\partial T}{\partial z} \right), ...
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2answers
82 views

Explain this multivariate differential identity

$$ \frac{\partial|\nabla\phi|^2}{\partial\phi}=-2\nabla\cdot\nabla\phi$$ I would very appreciate that you help me . Please do it in detail, I am quite not good at such problems. There is something ...
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3answers
126 views

Solving Laplace's equation on a domain with moving boundary

Consider a function $X(\xi,\nu)$, $2\pi$ periodic in $\xi$ satisfying $$\nabla^2 X = 0$$ in a domain $D$ with $\nabla = (\partial_{\xi},\partial_{\nu})$. If I know the values of $X$ on the boundary ...
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2answers
161 views

Python, numpy and complex functions (PDE's)

Update 4 I have almost given up on getting this right. This is the solution to the time-independent Schrodinger's equation, so the analytical solution is: $\psi(x,t) = \psi(x,0)e^{\frac{-iE ...
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55 views

Solving PDE with state and time dependent boundary conditions

I am interested in solving the following PDE (heat equation): $$\frac{\partial u}{\partial t} = \kappa \frac{\partial ^2 u}{\partial x^2}$$ In order to solve it, I discretize space uniformly into $N$ ...
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53 views

Solution to PDE with differential boundary conditions

I have the following equations $$ a_t(x,t)=1-a(x,t)b(x,t)^\gamma+D_1a_{xx}(x,t) $$ and $$ b_t(x,t)=\alpha(a(x,t)b(x,t)^\gamma -b(x,t))+D_2b(x,t)_xx $$ where $a,b:]0;4\pi[\times \mathrm{R}_+ ...
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65 views

parallel computatioan of a PDE in MATLAB

I want to solve a 1-D PDE $(\partial_{tt} + \alpha\partial_t)u(x,t)=\partial_{xx}u(x,t)-\sin(u(x,t))+f$, using method of lines and for this I defined a spatial grid of about n~1000 points. Since my ...
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1answer
134 views

How to project a vector into the H(div) space (in the context of finite elements)?

Say I have a simple elliptic PDE: $$ -\nabla\cdot(K\nabla p) = f \;\;\;\text{in}\;\Omega $$ with the appropriate boundary conditions. I solve for $p$ using a FEM (a discontinuous Galerkin method to ...
6
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1answer
62 views

Space-time finite element discretization for time-dependent PDEs

In FEM literature, semi-variational methods are typically used in the solution of time-dependent PDEs. I have not seen a fully-variational approach i.e. where space and time are discretised by FEM, ...
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1answer
132 views

Can the method of lines be used to discretize all PDEs?

I have found that the method of lines is a very natural way to think about the discretization of PDE's. Therefore I always default to that mindset when presented with a new set of equations. I have ...
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0answers
79 views

1 D Diffusion equation FDM with different layers

I'm trying to solve this particular equation $\frac{\partial u}{\partial t} = \frac{\partial}{\partial x} \big[D_{i}(x)\frac{\partial u}{\partial x} \big] + S(x,t)$ where the $i$ index denotes ...
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1answer
49 views

Reaction-Diffusion problem A->B, solving for B

I need to solve a Reaction-Diffusion using Finite Elements, piecewise linear elements. In this problem, a reaction $A \rightarrow B$, with rate law $ r_A = - k_A \cdot u_A $, takes part, where $u_i$ ...
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32 views

Isolated solving of coupled system of PDE [duplicate]

I would like to solve 3 differential equations for 3 unknowns. So I wrote a MATLAB code, which solves (using the '\' operator) these equation using a linear system of equations (in which the 3 ...
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2answers
126 views

Usability of upwind finite difference schemes

NOTE: I asked this on Mathematics Stack Exchange and there were no answers. So, I thought I might try here. Upwind schemes like the classic "upstream" scheme, can be used to solve, for example, the ...
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2answers
141 views

How do hexahedral FEM meshes improve approximation quality per degree of freedom, compared to tetrahredal meshes?

From the deal.II FAQ : ...quadrilaterals and hexahedra typically provide a significantly better approximation quality than triangular meshes with the same number of degrees of freedom; you ...
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0answers
89 views

numerical analysis of a partial integro-differential equation

I have to numerically solve a nonlinear partial integro-differential equation. This is my equation, $$\frac{\partial y(x,t)}{\partial t}=\int_{-\infty}^\infty K_0(|x-u|) \frac{\partial^2 ...
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3answers
157 views

Finite elements on manifold

I'd like to solve some PDEs on manifolds, say for example an elliptic equation on a sphere. Where do I start? I'd like to find something that use preexisting code/libraries in 2d , nothing so fancy ...
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1answer
165 views

Why are functional representations of systems important in numerical applications?

I tried asking a similar question in SE.Physics, and I got some information regarding the abstract side of this, but I figured I should post here to get more complete information about the numerical ...
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2answers
125 views

Well-posedness of a linear elasticity problem and Navier-Cauchy equation

I read a master thesis on a topic I'm interested too. This work concern the solution of the displacement equation of motion for a homogeneous, elastic, isotropic material: $$\rho \ddot{\mathbf{u}} - ...
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1answer
86 views

What does “strongly conservative” mean in the context of numerical methods?

I have a homework problem that asks me to show that 1st order unwinding or central differencing can give a strongly conservative, consistent scheme for the 1-D Burger's Equation using a finite volume ...
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3answers
145 views

Algorithms for radiation treatment planning

I have a medical physics problem - I want to maximise the dose absorbed by a brain tumour whilst minimising the dose in the rest of the brain, especially certain organs, such as the pituitary gland, ...
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3answers
122 views

Variable viscosity Stokes equation

One very efficient way to solve Stokes equation with periodic boundary conditions \begin{equation} -\eta \nabla^{2} \bf{v} + \nabla p = f \\ \nabla \cdot \bf{v} = 0 \end{equation} is using the ...
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0answers
144 views

Stationary 2D/3D Navier-Stokes source code

Trying to solve stationary Navier-Stokes problem for incompressible laminar Newtonian fluid. I've found a couple solutions for instationary Navier-Stokes equations (like FeniCS examples or CFD ...
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1answer
79 views

Minimization of The Blind Deconvolution Functional

I want to minimize the functional of teh Blind Deconvolution model as given in: Total Variation Blind Deconvolution by Chan and Wong. Their model is given by: $$ z = h \ast u + \eta $$ Where $ \ast ...
3
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2answers
250 views

The rate of convergence for finite difference methods for Poisson's equation with piecewise constant data

I am solving the following PDE; $$ \nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = \rho, $$ where $\rho(0.5,0.5) = 2$ (zero elsewhere), $0\leq x,y\leq1$ and the ...
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2answers
165 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 ...
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2answers
190 views

Neumann Boundary Condition at r=0 in Polar Coordinates (Numerical BCs)

I have asked a question in this regard earlier. I am trying to solve the following equation in Polar Co-ordinates: $$ u_t - (u_{rr} + \frac{1}{r} * u_r + \frac{1}{\theta} * u_{\theta\theta} + bu) = ...
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81 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 ...
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44 views

What is a reasonable manufactured solution to test finite difference method? [duplicate]

What is a reasonable manufactured solution to test the following equation against its finite difference approximation? I want it to look like a Cosine function about $0$, rotated about $Z$ axis ...
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1answer
45 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 ...
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1answer
41 views

Can you give some information for rothe method [closed]

I want to learn a numerical method for PDEs other than finite difference method. After some research on internet i have found Rothe method and it looks interesting to me. Unfortunately, i couldn't ...
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2answers
101 views

Numerical Solution of non-linear diffusion equation using Finite Differencing

I'm trying to solve the following non-linear diffusion equation: $$ \frac{\partial}{\partial t} u(x,t)= \frac{\partial^{2}}{\partial x^{2}}u(x,t)^{3}$$ $$ -1\leq x \leq1, t \geq 0 $$ with the boundary ...
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1answer
49 views

How can we compute the exact value of some rotational symmetric n-by-n convolution kernel

The commonly used $3\times 3$ Laplacian convolution kernel $\frac{1}{6}\begin{bmatrix}1 & 4 & 1\\4 & -20 & 4\\1 & 4 & 1\end{bmatrix}$ is only an approximation of rotational ...
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2answers
90 views

Can the conservative form of the advection equation be re-written by replacing the velocity term with an integral over all other points in space?

Suppose I have a 1D advection equation in conservation (divergence) form $\partial_t u(x,t) = -\partial_x [v(x)u(x,t)],$ where $u$ is a conserved quantity in space, and $v$ gives the velocity of the ...
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1answer
126 views

Inverse advection-diffusion problem, solving for a drift coefficient with experimental data?

I am investigating a physical process where I believe the 1-D advection-diffusion equation: \begin{equation} \frac{\partial u}{\partial t} = -\frac{\partial}{\partial x}[\mu(x,t) u(x,t)] + ...
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54 views

Frozen coefficient method (von Neumann stability analysis)

Earlier it was considered that frozen coefficients method for Neumann stability analysis for finite difference scheme is more heuristic than rigorous. But I have read some information in a book by ...
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1answer
163 views

What kind of test cases are convenient to use for testing the code for Euler equations of gas dynamics in polar coordinates?

Consider Euler equation of gas dynamics in polar coordinates as $$ \left( \begin{array}{ccc} \rho \\ \rho u_{r} \\ \rho u_{\theta} \\ E \end{array} \right)_{t} + \frac{1}{r} \left( ...
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223 views

MATLAB: incorrect computation with pdepe for parabolic-elliptic system

I have found that MATLAB solves 1D parabolic-elliptic system incorrectly by using pdepe function. Here is a system: $$ u_t = u_{xx} + 2, $$ $$ 0 = v_{xx} + u. $$ Boundary conditions: $$ ...
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1answer
159 views

Numerical method for solving PDE with non-linear boundary conditions

I have the following problem: \begin{align} \frac{\partial w}{\partial t} = \frac{1}{r^2} \frac{\partial}{\partial r}\left( r^2 D_1 \frac{\partial w}{\partial r} \right) \\ \\ \frac{dr_d}{dt} = ...
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Eigenvalue analysis of preconditioned partial differential operator

today, I encountered a confused problem by accident, but I have no ideas to deal with it fully. The question can be described as follows: for example, when we need to use FDM/FEM to discrete the ...
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2answers
74 views

Infinite Function Value on Dirichlet Boundary

I have been working on a multigrid solution to a non-homogeneous Dirichlet boundary value problem. However, the function goes to infinity on the boundary. This causes numerical overflow errors to be ...
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0answers
67 views

4th order tensor [closed]

I'm new with FEniCS and Python and I'm stuck with this issue: is there a way to write a 4th order tensor in an easy way to implement? I have to compute the following stiffnes tensor: $A_{ijkl}= ...
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1answer
56 views

What are the good testing problems for hyperbolic equation?

I read the whole list of this question: Where can one obtain good data sets/test problems for testing algorithms/routines? But the answers are in different areas and I want to ask a specific area. I ...
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2answers
85 views

Adaptive mesh refinement algorithms and the difference between AMR and moving mesh

I'm working on my thesis and a part of it has to do with adaptive mesh refinement. As a computer science major, I'm not too familiar with this field. The best way I can put my knowledge of AMR is: I ...
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53 views

Solvers for nonlinear parabolic PDEs [duplicate]

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 ...
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66 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 ...