The study of the propagation of errors in a numerical algorithm.
4
votes
1answer
51 views
Is stabilization of energy equation needed when momentum equation needs it?
When SUPG/PSPG stabilization is added to momentum equation of flow problem, is needed stabilization for energy equation also? I would guess that when stabilization for velocity works fine so one gets ...
1
vote
0answers
16 views
Simulation of asymmetric structures (occupancy = 0.5) unstable
I am trying to simulate a metal-organic framework in LAMMPS using the UFF potential. It's working quite well for some structures where all molecules have an occupancy of 1.
However, when I have a ...
5
votes
1answer
95 views
Stabilization of convection-dominated flow and turbulence modeling
Are stabilization techniques for convection-dominated flows like SUPG+PSPG, interior penalty methods, etc. able to handle turbulent flows without tubulence model being employed, at least up to some ...
3
votes
0answers
58 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} ...
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 ...
2
votes
2answers
124 views
Problem Condition and Algorithm Stability
Consider 2 mathematical problems:
$$
f_1(x) = a - x \\
f_2(x) = e^x -1
$$
The condition number for a function is defined as follows:
$$
k(f) = \left| x \cdot \frac{f'}{f} \right|
$$
Lets analyze ...
5
votes
3answers
122 views
What are the negatives of using higher order finite diference schemes?
I was looking at this wikipedia page:
http://en.wikipedia.org/wiki/Finite_difference_coefficient
It is a lists of higher order finite difference approximations, is there any negatives in using these ...
3
votes
1answer
77 views
Stability of numerical schemes for non-linear equations with a Jacobian with negative eigenvalues
Let us assume I have an A-stable numerical scheme. I believe that given any linear equation $y' = Ay$, it means that the numerical scheme applied to this equation is stable (and therefore convergent ...
4
votes
2answers
68 views
numerical investigation of stability of motion (confinement)
I am trying to find the required specifications of a RF trap, in which a proton can be confined.(trap dimensions,voltage frequency and amplitude used, etc). I have to solve the equations of motion ...
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 ...
2
votes
1answer
202 views
Meaning of CFL condition on parabolic problems
I've been studying this FEM theory and for the parabolic problems, there's the analysis of stability of the $\theta$-method.
I followed the analysis and they get this CFL (Courant-Friedrich-Lewy) ...
1
vote
1answer
60 views
Stability of forward euler method
I am trying to understand the stability of the forward Euler method. I read there's a model problem to see the stability.
$$y'(t) = \lambda y(t) \qquad t \in (0, \infty)$$
$$y(0) = 1$$
then the book ...
5
votes
2answers
54 views
Levenberg optimizer halts quickly when given more variables, or fewer constraints
I'm using the g2o C++ optimization library to refine a GPS trajectory using accelerometer data.
The program uses a Levenberg-Marquardt optimizer over data points representing the position and ...
4
votes
2answers
97 views
choice of the norm for Crank Nicolson stability estimate
I have a variable coefficient pde of the form $$u_t=c(t,x)u_{xx}, t\in [0,T], x\in [0,1]$$ with initial data $u_0=u(0,x)$ and $c(x,t)\in C([0,T]\times[0,1])$. I use three point discretization for the ...
3
votes
1answer
86 views
intuition behind the different discrete norms for Crank Nicolson
I am solving a heat equation $u_t=Au$ with Crank-Nicolson finite-difference method and $A$ is a usual discretization matrix for $u_{xx}$ term. I want to tell something about the whole error vector ...
1
vote
1answer
85 views
energy norm for transport equation
I asked this question before but did not have any luck with an answer. It might be a student level question but I need to understand that with possibly some help. I am considering the hyperbolic ...
5
votes
1answer
81 views
interpolation combined with methods of characteristics can cause oscillations for the transport equation?
I would like to know about the effect of using a higher order interpolator for the methods of characteristics. I am solving $$u_t+a(x,t)u_x=0$$ with some nonsmooth initial data $u_0(x)$ by the method ...
10
votes
1answer
159 views
Catastrophic cancellation in logsum
I'm trying to implement the following function in double-precision floating point with low relative error:
$$\mathrm{logsum}(x,y) = \log(\exp(x) + \exp(y))$$
This is used extensively in statistical ...
8
votes
2answers
244 views
Alternatives to von neumann stability analysis for finite difference methods
I'm working on solving the coupled one-dimensional poroelasticity equations (biot's model), given as:
$$-(\lambda+ 2\mu) \frac{\partial^2 u}{\partial x^2} + \frac{\partial p}{\partial x} = 0$$
...
5
votes
1answer
64 views
Linearized implicit time stepping
Consider the general FD implicit time stepping scheme
$\frac{x_{t+1} - x_t}{\Delta t} = f(x_{t+1})$,
where $x$ is the vector variable of interest and $f$ is some function, generally non-linear.
...
9
votes
3answers
117 views
Heuristic check of numerical stability
Assume I have a real valued function $f(x_1,\ldots ,x_N)$ of some variables $x_i$ which I want to evaluate numerically. In general the formula for $f$ can contain products, rationals, trancendental ...
4
votes
4answers
1k views
Estimating the Courant number for the Navier-Stokes Equations under differing Reynolds number regimes
I am familiar with the Courant-Friedrich-Lewy Condition in as far as it applies to the stability of explicit finite difference schemes for standard parabolic and hyperbolic PDEs. However, when ...
5
votes
3answers
215 views
analyze stability on a nonuniform grid
Assume you have a stability constraint between the space distance in time and space, for example, with an explicit Euler method for $u_t=u_{xx}$ we know $\tau\leq h^2/2$. That is, one can do stability ...
3
votes
1answer
81 views
Numerically stable real solution(s) to a system of bivariate quadratics
I have a a system of bivariate polynomials as follows:
$ E(u,v): e_2(u) v^2 + e_1(u) v + e_0(v) = 0 \\
F(u,v): f_2(u) v^2 + f_1(u) v + f_0(v) = 0$
where $e_n(u) = e_{n_2}u^2 + e_{n_1} u + e_{n_0}$ ...
4
votes
2answers
337 views
Solution oscillations with a small timestep in backward Euler
I am using backward Euler in a FEM scheme for a convection-diffusion problem. On a given mesh, I can take arbitrarily large time steps, as expected. But if I decrease time step, at some point it will ...
7
votes
2answers
165 views
Where can I find a good reference for the stability properties of several methods of solving parabolic PDEs?
Right now I have a code that uses the Crank-Nicholson algorithm, but I think that I would like to move to a higher-order algorithm for timestepping. I know that the Crank-Nicholson algorithm is stable ...
3
votes
2answers
152 views
Looking for a mathematical proof of stability in floating point arithmetic of CG - any reference?
I am looking for a reference - paper, book, discussion, anything that has a mathematical proof for stability of the conjugate gradient method in floating point arithmetic.
Something similar for ...
4
votes
1answer
144 views
Discretization of Classical Density Functional Theory (CDFT) problems
I formulate questions about ion densities in biological and materials problems using Classical Density Functional Theory, as in this paper which is also on the arXiv. The discretization in that paper ...
13
votes
2answers
437 views
How to determine if a numerical solution to a PDE is converging to a continuum solution?
The Lax equivalence theorem states that consistency and stability of a numerical scheme for a linear initial value problem is a necessary and sufficient condition for convergence. But for nonlinear ...
8
votes
1answer
180 views
What spatial discretizations work for incompressible flow with anisotropic boundary meshes?
High Reynolds number flows produce very thin boundary layers. If wall resolution is used in Large Eddy Simulation, the aspect ratio may be on the order of $10^6$. Many methods become unstable in this ...
10
votes
1answer
289 views
When should implicit methods be used in the integration of hyperbolic PDEs?
Numerical methods for solving PDEs (or ODEs) fall into two broad categories: explicit and implicit methods. Implicit methods allow larger stable timesteps but require more work per step. For ...
