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.
1
vote
0answers
24 views
Explicit 4th order space wave equation not stable implementation?
The explicit 4th order discretization for the 2D scalar wave equation is given by:
\begin{eqnarray}
U_{jk}^{n+1} = \left( \frac{\Delta t V_{jk} }{\Delta s} \right) ^2 \left( \sum_{a=-N}^N w_a ...
5
votes
1answer
45 views
Using fixed point iteration to decouple a system of pde's
Suppose I had a boundary value problem:
$$\frac{d^2u}{dx^2} + \frac{dv}{dx}=f \text{ in } \Omega$$
$$\frac{du}{dx} +\frac{d^2v}{dx^2} =g \text{ in } \Omega$$
$$u=h \text{ in } \partial\Omega$$
My ...
4
votes
1answer
49 views
Energy Conservation
I'm working on a time integration scheme for my research. As a result, I have come across an interesting phenomenon. Somehow, the total energy of the scheme oscillates. At any given time the total ...
2
votes
1answer
46 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 ...
2
votes
0answers
49 views
Von Newman stability analysis for 2D acoustic wave equation explicit
Von Newman stability analysis for acoustic wave equation explicit centered differences: 2nd order time and space (N 2)'th order:
\begin{eqnarray}
U_{jk}^{n+1} = \left( \frac{\Delta t V_{jk} ...
2
votes
0answers
67 views
Fenics:Result of Steady state dynamic linear elastic doesnt match with actual values
I solved the steady state dynamic linear elastic model in a solid. My equation is a function of frequency and the strong form is:
$$\operatorname{div}(\operatorname{stress}(\vec x, w)) + w^2 \rho u( ...
7
votes
2answers
86 views
Is the maximum/minimum principle of the heat equation maintained by the Crank-Nicolson discretization?
I'm using the Crank-Nicolson finite difference scheme to solve a 1D heat equation. I'm wondering if the maximum/minimum principle of the heat equation (i.e. that the maximum/minimum occurs at the ...
2
votes
1answer
40 views
Local truncation error and transformation of coordinates
I am given the advection equation
$$
u_t=u_x
$$
and then the transformation of coordinates
$$
x=x(\xi,\theta), \qquad t=\theta
$$
which leads us to the transformed equation
$$
x_{\xi} u_{\theta} - ...
5
votes
1answer
81 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 ...
12
votes
1answer
193 views
Convergence rate of FFT Poisson solver
What is the theoretical convergence rate for an FFT Poison solver?
I am solving a Poisson equation:
$$\nabla^2 V_H(x, y, z) = -4\pi n(x, y, z)$$
with
$$n(x, y, z) = {3\over\pi} ((x-1)^2 + (y-1)^2 + ...
9
votes
2answers
99 views
What is pseudo time-stepping?
While reading some literature on PDE solvers I came across the term pseudo time-stepping today. It seems to be a common term, however I failed to find a good definition or an introductionary article ...
5
votes
1answer
158 views
Implementing Explicit formulation of 1D wave equation in Matlab
So the theory is straightforward. We have:
$$\frac{\partial^2U}{\partial t^2}=c^2 \frac{\partial^2U}{\partial x^2}$$
discretizing it gives:
$$\frac{U(i+1,j)- 2U(i,j) + U(i-1,j)}{(\Delta t)^2} = c^2 ...
3
votes
1answer
54 views
How to perform multigrid technique when relaxation methods don't converge?
It is well known that, when a system of linear equations is obtained from discretization of partial differential equation, the solution process can be accelerate significantly by multigrid technique. ...
2
votes
1answer
103 views
Role of boundary conditions (e.g. periodic) in Poisson equation
Given 3D Poisson equation
$$
\nabla^2 \phi(x, y, z) = f(x, y, z)
$$
and the right hand side and the domain, am I free to impose any boundary conditions (BC) on the function $\phi$, or do they have to ...
5
votes
0answers
109 views
frozen coefficient vs. constant coefficient
This is a follow up to the question about the method of frozen coefficients. I think it deserves to be a separate question. The frozen coefficient problems are obtained by fixing the coefficients of ...
4
votes
0answers
85 views
method of frozen coefficients and its relation to the von Neumann stability analysis
I am considering two equations $$u_t=a(x)u_{xx}$$ and $$v_t=b(x)v_x$$ as classical representatives of the parabolic and hyperbolic family of equations. If $a(x)=a$ and $b(x)=b$ were constants, to show ...
1
vote
1answer
61 views
Von Neumann Stability Analysis
I came across the following task recently:
Use the von-Neumann stability analysis to investigate the stability of the discrete form of $\frac{\partial c}{\partial x} = \frac{\partial^2 c}{\partial ...
1
vote
1answer
76 views
4th order Padé scheme formula derivation
I am trying to derive the formula of the 4th order Padé scheme that passes through the points $x_i$, $x_{i-1}$ and $x_{i+1}$
$$\Big(\frac{\partial\phi}{\partial x} \Big)_i = ...
2
votes
1answer
58 views
Location of Unknowns in Unstructured Mesh
I am currently learning a code which utilizes Scharfetter-Gummel discretization for unsteady drift-diffusion equations. For this scheme, a 2D unstructured triangular mesh is used, with the unknowns ...
6
votes
4answers
167 views
Reference request: Rigorous analysis of algorithms for PDE and ODE
I'm interested in suggestions for book references on the subject of numerical PDE and ODE, in particular, a rigorous analysis of such methods in a manner written for professional mathematicians. It ...
4
votes
2answers
129 views
No flux boundaries for mixed hyperbolic parabolic PDE
I read this post, "Conservation of a physical quantity when using Neumann boundary conditions applied to the advection-diffusion equation" and although it is the same type of equation it does not fit ...
5
votes
2answers
96 views
Are the drift-diffusion equations from semiconductor physics analogous to solving an advection-diffusion problem?
I am trying to understand an extra terms that appears when I derive the drift-diffusion equations for semiconductors. The extra term (see below) comes from applying the chain rule to the advection ...
1
vote
0answers
31 views
Can I use RANS to see the effect of mixed convection?
my question is: can I use a RANS simulation to see the effect of mixed/natural convection?
Actually I have also a second question: I would like to do this in Comsol multiphysics, but it seems that it ...
2
votes
3answers
106 views
Open boundary conditions with the advection-diffusion equation
Following on from my previous equation I'm would like to apply open boundary condition to the advection-diffusion equation (with reaction term),
$$ \frac{\partial \phi}{\partial t} = ...
0
votes
0answers
66 views
What's the best programming language to learn for solving partial differential equations? [closed]
I have to create a program that compares two or three
different methods (FEM FVM FDM) for solving an easy pde.
Is there a program language in which I could do this easily?
Thank you
1
vote
0answers
32 views
Nondimensionalization of time using time-dependent variable
I'm not sure if it is appropriate to put post the question here, but considering that nondimensionalization is an important step in solving pde numerically.
Suppose from experiments I have two rates ...
6
votes
1answer
158 views
Conservation of a physical quantity when using Neumann boundary conditions applied to the advection-diffusion equation
I don't understand the different behaviour of the advection-diffusion equation when I apply different boundary conditions. My motivation is the simulation of a real physical quantity (particle ...
12
votes
1answer
220 views
Strange oscillation when solving the advection equation by finite-difference with fully closed Neumann boundary conditions (reflection at boundaries)
I am trying to solving the advection equation but have a strange oscillation appearing in the solution when the wave reflects from the boundaries. If anybody has seen this artefact before I would be ...
10
votes
2answers
243 views
Is Crank-Nicolson a stable discretization scheme for Reaction-Diffusion-Advection (convection) equation?
I am not very familiar with the common discretization schemes for PDEs. I know that Crank-Nicolson is popular scheme for discretizing the diffusion equation. Is also a good choice for the advection ...
4
votes
1answer
113 views
mathematical statement of “open” boundary condition
For your information, the original equation comes from here. Note: You DON'T have to read the paper. I will make the question as self-contained as possible.
The central equation to solve is equation ...
2
votes
1answer
113 views
FEM for non-divergence form elliptic equation
The FEM is usually used with a weak form of PDE. But for the non-divergence form elliptic operator
$$
-a_1(x,y) \frac{\partial^2}{\partial x^2} - a_2(x,y) \frac{\partial^2}{\partial y^2}
$$
or ...
4
votes
2answers
87 views
Regularization of a discontinuous source term in an elliptic pde
Suppose I'm solving $$\frac{d}{dx}\left(K(x)\frac{du}{dx}\right)=f \text{ in }\Omega,$$ $$u=g \text{ on } \partial\Omega$$where $K(x)$ is smooth and
$$
f(x) = \left\{
\begin{array}{ll}
...
5
votes
0answers
79 views
Understanding and implementing the Heterogeneous Multiscale Finite Element Method
I'm following the explanation given by Weinan E and Bjorn Engquist (1994), pp 26-29, and I have a few questions about it. To understand my questions, I'll first try to explain what think I know, and ...
0
votes
0answers
52 views
Modified heat eq. in 3D
please how can I write gradient numerically?
I have an eq.
$\dfrac{\partial X}{\partial t}=\nabla^2 X + u\nabla X$
I wonder how write this in 3D.
Many thanks for any idea... I can find how to solve ...
2
votes
3answers
194 views
necessary and sufficient tests to show order of convergence for the numerical method
I would like to know what are the necessary and sufficient tests one has to perform in order to show the convergence of the algorithm. I have not found a good reference to state for that as I am ...
3
votes
1answer
77 views
Iterative Block Matrix Splitting for Multiphysics Simulation
I have a problem of the form
$$\left[\begin{array}{cc}
-(\lambda+2\mu)\frac{d^2}{dx^2} & \alpha\frac{d}{dx} \\
\frac{\alpha}{\Delta t}\frac{d}{dx} & \frac{c_0}{\Delta ...
0
votes
0answers
40 views
boundary condition impact on the Fourier stability analysis
I am looking for some reference on the stability analysis of the finite difference scheme for the linear constant coefficient pde. I have a few books and I see how the Fourier analysis is used but ...
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
...
6
votes
2answers
144 views
discrete $L^p$ norms for non-uniform grid
I am reading a book on numerical methods and the square of the discrete $L^2$ norm is defined as $$||x||^2_2=h\sum_1^Nx^2_i$$
Every point gets a "weight", which is $h$, thus this is like an average ...
3
votes
1answer
105 views
A better Fast Marching Method?
I am using the Fast Marching Method (FMM) to calculate shortest "distance" (traveltime) from some points.
The way FMM works is: I keep a velocity function in RAM: V(xi,yj,zk). I also keep a priority ...
5
votes
1answer
123 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 ...
5
votes
1answer
252 views
How to properly apply non-homogeneous Dirichlet boundary conditions with FEM?
In general, Dirichlet boundary conditions won't be satisfied exactly for FEM for non-homogeneous boundary conditions. The FEM codes I've seen set the degrees of freedom to interpolate the Dirichlet ...
2
votes
1answer
79 views
Pointwise convergence
I have seen a number of papers that propose a finite-difference method and then show the numerical results for it. Without providing a rigorous analysis(can be some summary or note or whatever, just ...
0
votes
0answers
135 views
solving nonlinear differential equations by finite differences+matlab code [closed]
Here is the equation that I don't know how to solve by finite differences. I will appreciate when someone can help me.
$$
\frac{\partial{^2T}}{\partial{x^2}} = 0.01 \cdot (T-20)^4 \\
T(0) = 200 \\
...
2
votes
0answers
135 views
Solving PDE or eigenvalue problems without FEM
Do you know any methods/solvers for PDE or eigenvalue problems like
$\begin{cases} \Delta u= 0\ (\text{ or }\lambda u) & \text{ in }\Omega \\ u =0 & \text{ on }\partial \Omega \end{cases}$ ...
0
votes
0answers
34 views
Boundary Conditions in Heat Transfer Finite Element Method
I'm using finite elements method for solving Heat-Transfer problem in 3D space. And there is a problem with 3rd and 2nd type boundary conditions: there are some oscillations near boundaries (in the ...
4
votes
0answers
56 views
Solving diffusion PDE using finite differences
I need some hints on how to solve this diffusion equation ($\alpha, k_1,k_2$ and $k_3$ are constants):
$$ {\partial P \over \partial y} + k_1 {\partial P \over \partial t} + \alpha P = {1 \over k_2} ...
6
votes
3answers
215 views
multigrid method to solve PDE
I need simple explanation of the Multigrid Method or some literature about this.
I am familiar with iterational methods including BiCGStab,CG,GS,Jacobi and preconditioning, but I am a beginner with ...
0
votes
1answer
50 views
multigrid method to solve PDE [duplicate]
Possible Duplicate:
multigrid method to solve PDE
I need explanation of the Multigrid Method or some literature.
I am familiar with iterational methods including BiCGStab,CG,GS,Jacobi and ...
1
vote
0answers
83 views
Numerics for heat equation
I want to simulate on my computer the solution of the heat equation in 3 space dimensions with Cauchy initial data, that is
$$\partial_t u=Tr[A(x)\cdot \Delta u], u(0,x)=u_0(x) $$
where $u_0\in ...
