Tagged Questions
3
votes
0answers
26 views
Timestepping PDE with positive eigenvalues
I'm trying to numerically solve a PDE, namely:
$$
\partial_t \binom{u(x, t)}{v(x, t)}
= x \left(\begin{array}{cc}
0 & -1 \\
1 & 0
\end{array}\right) \cdot
\partial_x \binom{u(x, t)}{v(x, t)}
...
0
votes
0answers
14 views
Finite-Element Method: Question on stability
I am trying to determine the stability of the PDE
http://mathurl.com/cazterh
given the finite-element scheme
http://mathurl.com/cetadmr
and constant s
http://mathurl.com/bcfq5us
My problem is ...
0
votes
0answers
8 views
Reference request: Finite difference methods on curvilinear (body fitted) grids
I was wondering if someone may be aware of some form of detailed summary (book, tutorial paper) about the use of finite difference methods on curvilinear (body fitted) grids.
I was only able to ...
0
votes
0answers
17 views
Von Newman stability analysis for 2D acoustic wave equation explicit
Von Newman stability analysis for acoustic wave equation explicit centered differences: 2nd order time and space (N 2)'th order:
\begin{eqnarray}
U_{jk}^{n+1} = \left( \frac{\Delta t V_{jk} ...
0
votes
1answer
29 views
solve a “wave equation” with an extra term
I want to solve the following "wave equation" $$\nabla^2\psi(\vec{r},t) - \frac{1}{c(\vec{r})^2}\frac{\partial^2}{\partial t^2}\psi(\vec{r},t) = R(r)\psi(\vec{r},t)$$ subject to initial conditions ...
3
votes
1answer
53 views
Finite difference implicit schema for wave equation 1d not unconditionally stable?
The wave equation 1D with constant density is defined as:
\begin{equation}
\frac{\partial^2 U}{\partial t ^2} = V^2 \frac{\partial^2 U}{\partial x ^2}
\label{eqa}
\end{equation}
And the implicit ...
2
votes
1answer
63 views
Is the backward Euler method A-stable for non-linear equation?
I'm looking at the parabolic equation
$$
\frac{\partial u}{\partial t} - \frac{\partial}{\partial x}\left(\alpha(u)\frac{\partial u}{\partial x}\right) = f(x,t)
$$
I know the Backwards Euler method ...
0
votes
1answer
90 views
Two Dimension Heat Equation ADI Local Truncation Error
Given a two dimensional heat equation $\displaystyle \frac{\partial u}{\partial t}=K(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2})$ solved using ADI (the alternating-direction ...
1
vote
1answer
115 views
Von Neumann Stability Analysis
I came across the following task recently:
Use the von-Neumann stability analysis to investigate the stability of the discrete form of $\frac{\partial c}{\partial x} = \frac{\partial^2 c}{\partial ...
1
vote
1answer
140 views
Variational formulation of Robin boundary value problem for Poisson equation in finite element methods
So I am confused about some details of obtaining a variational formulation specifically for Poisson's equation. I am in a Scientific Computing class and we just started discussing FEM for Poisson's ...
0
votes
1answer
81 views
finite differences for PDE's
I am having trouble with this question, much appreciated if anyone can help?
a) for the 2nd order wave equation:
\begin{align}
&\partial_{tt}u(x,t)-c^2\partial_{xx}u(x,t)=0 & (x,t) \in ...
4
votes
0answers
162 views
How can I solve the Poisson PDE efficiently and fast in cylindrical coordinates?
I am trying to numerically solve the Possion PDE in cylindrical coordinate system.
$$\Delta f = {1 \over \rho} {\partial \over \partial \rho} \left(\rho {\partial f \over \partial \rho} \right) + {1 ...
1
vote
0answers
28 views
How to can I transform the 2D cuasi Laplace equation with variable coefficients to finite difference scheme?
I want to solve $$\frac{\partial}{\partial x}\left(\frac{1}{\rho(x,y)}\frac{\partial \Phi}{\partial x}\right) + \frac{\partial}{\partial y}\left(\frac{1}{\rho(x,y)}\frac{\partial \Phi}{\partial ...
0
votes
0answers
19 views
characteristics of equation $u_t + a u _ x = 0$
Sketch the characteristics for the equation $u_t + a u _ x = 0$ for
$0 \leq x \leq 1$ when $a = a(x)= x-1/2$.
Set up the upwind scheme on a uniform mesh, $x_j = j \Delta x,
j = 0, 1, \cdots , J$, ...
0
votes
2answers
84 views
need an example of 2D wave equations with analytic solutions
I need to numerically solve the following wave equation$$\nabla^2\psi(\vec{r},t) - \frac{1}{c(\vec{r})^2}\frac{\partial^2}{\partial t^2}\psi(\vec{r},t) = -s(\vec{r},t)$$ subject to zero initial ...
0
votes
0answers
29 views
Five points discretization
Show that the standard 5-points discretization (FDM) of the Laplace operator provides a boundary maximum principle.
How I prove that!!
0
votes
0answers
26 views
Discretize the Laplacian operator(FDM)
How to solve this question please!!
Discretize the Laplacian operator $∆u$ in two dimensions by a full compact scheme, i.e
$∆ ̃_h u(x,y)≔1/h^2 [ α u(x,y)+ β (u(x-h,y)+u(x+h,y)+u(x,y-h)+u(x,y+h) )+γ ...
2
votes
1answer
61 views
General quasilinear PDE - derivation of characteristic equation
A general inhomogeneous quasilinear PDE is given as
$a(x,t,u)u_t + b(x,t,u)u_x = c(x,t,u)$.
In the derivation of the characteristic equations it says one can consider the solution to this PDE as the ...
0
votes
2answers
60 views
Numerical integration of a derivative dataset
I have an experimentally measured derivative data ($\frac {dy}{dx}$) at a range of times i.e $\frac {dy}{dx}$ for $0 \le t \le tf$.
Integrating for $t=0$ is fine, since $y(x,0)$ is known. However, ...
0
votes
1answer
40 views
How can I modify the Eikonal equation to have smooth iso-contours?
I am solving the Eikonal Equation in 2D:
$ | \nabla T(x,y)|=1/V(x,y) $
for the traveltime, T(x,y), from a starting point:
$ T(x_0,y_0) = 0$.
The curves $ T(x,y)=C$ forms closed contours around the ...
5
votes
1answer
74 views
How should a numerical solver treat conserved quantities?
Since the good old Noether Theorem ultimately states that any physical system will exhibit conserved quantities, how should they be treated best in numerical solvers?
On the one hand, one can observe ...
3
votes
1answer
113 views
Weak Formulations and Lax Milgram:
I have a question on how to put a PDE into weak form, and more importantly, how to properly choose the space of test functions. I know that for an elliptic problem, we want to start with a problem ...
1
vote
1answer
83 views
Derivation of weak form for variational problem
My question is about understanding the derivation of the weak form of a variational problem (to be used for the solution via the finite element method).
The problem is as follows (it is an image ...
1
vote
0answers
59 views
Reference for Finite Difference Schemes
Is there any place that I can find a list of different PDEs and common finite difference schemes used for each? I have seen tables of finite difference coefficients such as the one here ...
1
vote
0answers
33 views
Characteristic-Galerkin convergence rate
I am reading the following article:
Characteristic-Galerkin and Galerkin/least-squares space-time formulations for the advectiondiffusion equation with time-dependent domains
by O. Pironneau, et al.
...
2
votes
1answer
51 views
Local truncation error for the forward-difference method
I need to show that if $\gamma=\frac{K\tau}{h^2}=\frac{1}{6}$, then in the explicit forward-difference method $\frac{w_{k,j+1}-w_{k,j}}{\tau}-K\frac{w_{k+1,j}-2w_{k,j}-w_{k-1,j}}{h^2}=0$ the local ...
2
votes
1answer
98 views
Why are hyperbolic pdes considered difficult to solve numerically?
I am asking only out of curiosity, I did sufficient fluid mechanics to be familiar with shockwaves and the such, but I'm wondering how it relates to the numerical qualities of hyperbolic pdes in ...
4
votes
1answer
76 views
Stability of pde in some $L^p$ norm and stability of a numerical scheme for it equivalence.
I would like to get some light on how to proceed and my confusion. I consider some IBVP of the form $$u_t+L(t,x)u=0, x\in D, t\in [0,T]$$ with some BC and initial data. And I use some numerical method ...
2
votes
0answers
50 views
Initial Conditions for Finite Difference of PDE
I am having trouble with figuring out what my initial conditions should be for a simple finite difference algorithm I wrote in Matlab.
Specifically, I'm trying to show that the regular 1-Dimensional ...
1
vote
0answers
81 views
Transient state on heat diffusion equation numerical solution
I'm trying to find a transient temperature of a certain location of a 3D body, after a known perturbation. I'm solving the heat equation using finite differences. I've tried the explicit, implicit and ...
0
votes
0answers
54 views
How does one more Efficiently Numerically Solve Multidimensional Problems using Spectral Methods?
As per the title, would you please tell me how to more efficiently solve multidimensional partial differential equations? Other then just tediously writing out the matrix elements manually. If you ...
1
vote
0answers
55 views
Orthogonal vs general curvilinear coordinates
Solutions to PDEs over irregular domains can be computed using the finite difference method by the introduction of so called body fitted coordinate systems where the coordinate lines are aligned to ...
0
votes
0answers
448 views
How to solve these 2-D coupled differential equations in Matlab?
I have three partial differential equations (PDEs) and an analytical solution for a variable as shown. Using these equations I want to solve for \phi(x,y,t), p(x,y,t), C_{a}(x,y,t) and C_{b}(x,y,t) ...
0
votes
0answers
89 views
Numerical solution for a system of PDEs via steady state with time integration method
I'm working to solve the steady-state short circuit current of a solar cells, using the coupled continuity equations with a drift-diffusion expression and Poisson's equation:
D[n[x, t], t] - (1 / q) ...
2
votes
0answers
118 views
Helmholtz and Biharmonic equation examples with exact solution
I'm looking for examples of Helmholtz and Biharmonic equations in Cartesian co-ordinates with exact solutions, in order to compare results of my numerical solutions with it.
I was able to find quite ...
1
vote
1answer
59 views
Numerical Computation of Eigenvalues
I am trying to find the first few eigenvalues of an operator defined by the following PDE:
$$ \begin{cases} -\Delta u +(1-\varphi)u=\lambda u, & \text{ on }\Omega = [0,1]^2 \\
u=0 & \text{ ...
1
vote
0answers
177 views
Stability of Lax-Wendroff scheme for wave equation
I'm a beginner in PDE's and numerical methods, so please go slow :-D
I'm trying to show that the Lax-Wendroff scheme is stable for $|a\lambda| < 1$.
The scheme is this:
$v_m^{n+1} = v_m^n - ...
1
vote
1answer
69 views
Magnus expansion for linear operators
I want to learn about Magnus expansion for the time dependent pdes of the form $u_t(t,x)=A(t)u(t,x)$. According to the wikipedia explanation http://en.wikipedia.org/wiki/Magnus_expansion $A$ has to be ...
0
votes
1answer
76 views
Multigrid tutorial/book
I was reading Press et. al., "Numerical Recipes" book, which contain section about multigrid method for numerically solving boundary value problems.
However, the chapter is quite brief and I would ...
1
vote
2answers
163 views
how to solve kawahara equation?
the numerical solution to a problem involving a nonlinear partial differential equation of the form below
$u_t + uu_x + u_{3x} − u_{5x} = 0$
$u(x, 0) = f (x) , x ∈ R$
which is called Kawahara ...
0
votes
0answers
34 views
Proof of convergence Finite Volume
Hi all can some one give me a good reference to the proof of convergence and error estimation of ELLIPTIC PDE in divergence form by a Cell centered Finite Volume Method..???
Arwin
2
votes
0answers
133 views
Fourier transform for Neumann boundary condition
I need to solve system of two coupled partial differential equations numerically.
$\frac{\partial x_1}{\partial t} = c_1\nabla ^2 x_1 + f_1(x_1,x_2) \\$
$\frac{\partial x_2}{\partial t} = ...
1
vote
2answers
359 views
Fourier transform of heat equation
I need to solve following partial differential equation with Fourier transform numerically.
$
\frac{\partial T}{\partial t} = \nabla(c\nabla T)
$
where T is temperature, c heat conductivity and t is ...
0
votes
1answer
127 views
Numerical Solution of Systems of PDE
Could someone give me some reference to the Numerical solution of a System of PDEs of following type.. (which also encompasses strongly elliptic system of PDEs) in 2D or 3D.
$$\left\{
...
1
vote
1answer
48 views
Finite element method - Dual functional Error estimate
I have an equation -$u''=1,~~x\in(0,1)$. I solved it numerically by Finite element method. and find an approximate solution. $u_{h}$. As you know to do this I defined bilinear and linear functionals ...
2
votes
1answer
435 views
Solution to a second order semilinear elliptic PDE
I am trying to find a solution (if one exists) to this PDE where (r,z) are cylindrical coordinates. $$u_{rr}+u_{zz}+u_{r}/r-u/r^2 = u^2u_{r}/r+u(u_{r})^2+u(u_z)^2$$ u is a dimensionless function ...
0
votes
1answer
74 views
stability definition for nonhomogeneous pde
I see the definition for stability for any solution operator $E$ is that $$||Eu^{n}||^2=||u^{n+1}||^2\leq C||u^{n}||^2$$ for some constant $C$ and some pde $u_t=Lu$. However, I can show that ...
4
votes
1answer
163 views
How do I numerically calculate a function from its noisy gradient using “global integration”?
I have the model $\ s(x,y)=x^2+y^2, 0 \leq x \leq 1, 0 \leq y \leq 1 $.
Instead of observing the model directly I am observing the derivatives of the model + some noise (e):
$\ p(x,y)=s_x+e, ...
1
vote
1answer
120 views
How do I combine two matrix equations into one?
I have a discretely sampled 2D function:
S =
1 2 3 4
1 2 3 4
1 2 3 4
I want to find finite difference ...
0
votes
0answers
150 views
Least Squares solution of linear equation using conjugate gradient method
I have two partial differential equations that I want to solve (for s) by finite differences:
$\begin{align*} \frac{\partial}{\partial x}s(x,y) &= p(x,y)\\
\frac{\partial}{\partial y}s(x,y) ...
