The study of the propagation of errors in a numerical algorithm.

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2
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2answers
73 views

Lax-Richtmyer stability analysis

I would like to get to know more in details about Lax-Richtmyer stability analysis (esp in examples), but I didn't manage to find anything except a definition. Could you advice any sources for this ...
6
votes
3answers
203 views

Finite Difference Method Stability

The diffusion equation is: $ \frac{\partial T}{\partial t} = \alpha \left( \frac{\partial^2 T}{\partial x^2} \right) $ An explicit finite difference approach can be used to solve this, forward in ...
1
vote
0answers
20 views

Error analysis and the Model Problem [closed]

In numerical methods for ODE's, the model problem y' = cy where c is complex is regarded as sufficient in performing error analysis for different methods in ...
2
votes
1answer
116 views

Von Neumann stability analysis in 3d

I need to get a stability criterion for the numerical scheme for equation $$\frac{\partial u}{\partial t}-\frac{\partial^2 u}{\partial x^2}-\frac{\partial^2 u}{\partial y^2}-\frac{\partial^2 ...
2
votes
1answer
116 views

Courant Friedrichs Lewy condition - how to get it?

I am interested, how can we get CFL condition for every type of PDE? It's known that for 1st order linear equation $$\frac{\partial u}{\partial t}+a\frac{\partial u}{\partial x}=0 $$ CFL is get from ...
2
votes
2answers
150 views

In what regime do the continuous and discontinuous Galerkin method become unstable for advection-diffusion systems?

I know that the finite volume method (based around a central different stencil) is unstable for advection dominated advection-diffusion problems. This leads to different adaptive schemes to can be ...
4
votes
1answer
101 views

Von Neumann stability analysis of coupled equations

Given a forward-in-time approximation I have the coupled equations: $$ \frac{T^{(n+1)} - T^{(n)}}{\Delta t} = x T^{(n)} - y h^{(n)} \\ \frac{h^{(n+1)} - h^{(n)}}{\Delta t} = -z h^{(n)} - \alpha ...
1
vote
0answers
57 views

Stability question (finite difference): dealing with corner nodes

Consider one initial boundary value problem for sphere. $$\frac{\partial u}{\partial t}=\operatorname{div}A\nabla u +f$$ Here is explicit numerical scheme (we consider that it is stable): ...
5
votes
1answer
201 views

Stable time step limits for Velocity-Verlet integration

I'm implementing a mass-spring solid mechanics solver and I'd like to use the Velocity-Verlet time integration scheme. However, I cannot find anything about the maximum stable time step -- either ...
5
votes
2answers
132 views

Stability of the first-order exponential integrator method

The question is about the first-order exponential integration method described in this article. Consider a system of ordinary differential equations $$y'(t) = -A\,y(t) + \mathcal{N}(t, y), \qquad ...
2
votes
3answers
197 views

Caculating the mean of vector accurately

I am having trouble with calculating a mean of vector with sufficient accuracy. My current solution which works but it quite slow and has unpredictable performance: mean_sum = mean = ...
3
votes
1answer
398 views

Stability of numerical method for 1D Burger's equation

I am trying to solve 1D viscous Burger's equation numerically and I cannot apply von Neumann analysis because the equation is non-linear. How do I predict the stability criteria for my system? I also ...
4
votes
2answers
187 views

Numerical instability of spherical pendulum

Problem statement I am trying to simulate a spherical pendulum, with rod length $r$ south-polar angle $\theta$ and azimuthal angle $\phi$ initial values $(\theta_0,\phi_0)= (0,0)$ My particular ...
4
votes
1answer
335 views

Finite elements for Stokes with traction boundary conditions

Suppose we are given the Stokes equations with Neumann conditions on part of the boundary: $-\nabla\cdot\boldsymbol{\sigma} = \mathbf{f}, \quad \text{and} \quad \nabla\cdot \mathbf{u} = 0 \quad ...
2
votes
1answer
47 views

Model Spinup time

While I look up programming and technical stuff on Stack Overflow all the time, I figured that would probably be the wrong forum for a question like this. Some background: I'm a climate modeler ...
4
votes
2answers
179 views

How to decide stability of Runge-Kutta method for non-linear ODE?

I'm working on a parameter study of Duffing's equation $\ddot x + \delta \dot x + \alpha x + \beta x^3 = \gamma \cos{\omega t},$ where $\delta, \alpha, \beta, \gamma$ and $\omega$ are real ...
5
votes
1answer
137 views

Stabilization of solution to one-dimensional system of PDE

I am trying to solve numerically next PDE system: $$\frac{\partial c}{\partial t}=\epsilon\frac{\partial}{\partial x}(\frac{\partial c}{\partial x}+\rho\frac{\partial \varphi}{\partial ...
4
votes
1answer
76 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
28 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 ...
6
votes
1answer
153 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 ...
4
votes
1answer
308 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
209 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 ...
4
votes
2answers
151 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
154 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
213 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 ...
6
votes
2answers
133 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 ...
2
votes
1answer
651 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
155 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
92 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
134 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
117 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
121 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 ...
6
votes
1answer
116 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 ...
11
votes
1answer
182 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 ...
11
votes
2answers
501 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: $$-(\lambda+ 2\mu) \frac{\partial^2 u}{\partial x^2} + \frac{\partial p}{\partial x} = 0$$ ...
6
votes
1answer
126 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. ...
12
votes
3answers
174 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 ...
3
votes
4answers
3k 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
303 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
88 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
488 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 ...
8
votes
2answers
212 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
166 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
164 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
953 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 ...
11
votes
1answer
288 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 ...
11
votes
1answer
609 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 ...