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

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-1
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2answers
67 views

Stability of matrix equations in MATLAB

Is there a method to check for unconditional stability or positive-definiteness of large matrices in MATLAB? For example, I know that matrices with property M (positive main diagonal elements and ...
5
votes
0answers
81 views

Using entropy functions for increasing numerical stability

Regarding the numerical stabilization of two-dimensional advection equation, \begin{equation} \dfrac{\partial f}{\partial t} + \Big(\dfrac{d\varepsilon_1(k)}{dk}\Big)\dfrac{\partial f}{\partial z} - ...
3
votes
1answer
98 views

Stability Criterion for this Explicit Scheme

I am solving an unsteady flow using the dual-time Navier-Stokes equation in which I write my momentum equation as: $$\frac{\partial u}{\partial \tau} + \frac{\partial u}{\partial t} + \frac{\partial ...
-1
votes
1answer
77 views

Stability analysis for coupled nonlinear system of partial differential equations

I'm trying to solve a nonlinear partial differential equation \begin{equation} L(u_{xxtt},u_{xx}u_{tt},u_{xt}^2,u_{xt},u_{tt})=0 \end{equation} using finite difference methods. In order to remove the ...
1
vote
1answer
50 views

Calculate inverse of dense matrix with entries of very different magnitude

I need to calculate the inverse of a dense matrix, with some elements taking values as high as 1e9 and some around 1e2. What would be the best method to do it? Note: I am more concerned about the ...
6
votes
1answer
161 views

What does the Von Neumann's stability analysis tell us about non-linear finite difference equations?

I am reading a paper [1] where they solve the following non-linear equation \begin{equation} u_t + u_x + uu_x - u_{xxt} = 0 \end{equation} using finite difference methods. They also analyse the ...
13
votes
0answers
95 views

Usefulness of elements with mesh-dependent stability

After doing some mathematics related to the stability of elements in 3D Stokes problem I was slightly shocked to realize that $P_2-P_1$ is not stable for an arbitrary tetrahedral mesh. More precisely, ...
1
vote
0answers
27 views

Stability in Discretization of 1D Stationary Boltzmann equation

I want to discretize and numerically solve the following PDE: \begin{equation} v(k)\dfrac{\partial f}{\partial x} + E(x)\dfrac{\partial f}{\partial k} = S\{f\} \end{equation} using finite volume (box ...
3
votes
1answer
85 views

Stability in discretization of a PDE

Suppose I want to numerically solve for $f(x,k)$ the one-dimensional Boltzmann equation for electrons in steady-state condition, given as: \begin{equation} \left( \dfrac{\hbar k}{m} \right) ...
0
votes
1answer
77 views

The condition for stability using the leapfrog method

I have the ODE below $$\frac{d}{dt}\pmatrix{x\\ y} = \pmatrix{0 &1\\-a &0}\pmatrix{x\\ y} \enspace .$$ The $m=1$ leapfrog method is defined as: $$y_{n+1} = y_{n-1} + 2f_nh \enspace .$$ For ...
0
votes
1answer
53 views

How do we find the condition?

Suppose that we are given a numerical scheme. In order to find the CFL condition , we set $U_j^n= \lambda ^ne^{ik x_j}$ and put it into the numerical scheme. I have shown that the given method is ...
3
votes
0answers
136 views

CFL Condition and Convection Diffusion Equation in 2D

I am solving the convection-diffusion equation in 2D using Finite Differences with the $\theta$ scheme. The velocity of the fluid and the diffusion coefficient is low in my case (in the range of ...
0
votes
0answers
91 views

von Neumann stability analysis system of equations

How can I apply von Neumann stability analysis for the following system of equations? Discritization method: Lax method \begin{align} &\frac{\partial u}{\partial t} + a \frac{\partial v}{\partial ...
1
vote
2answers
107 views

significance of energy equation and its conservation

I am interested to know the significance of the energy conservation laws when modelling fluids (or other materials). Am I correct in saying that if energy is conserved then stability is achieved. In ...
2
votes
1answer
59 views

How can I analyze the stability of a PDE discretization at a boundary?

I have a numerical discretization of a partial differential equation that seems to be unstable or stable at a boundary point, depending on what finite difference scheme I am using. Are there standard ...
0
votes
0answers
90 views

Computing solutions with singularities using MATLAB ODE45

I am new to solving numerically ODES and thus it is difficult for me to judge the reliability/trustworthiness of the results that I have produced for the following problem: I am dealing with a 2nd ...
3
votes
2answers
180 views

Does artifical dissipation term makes scheme inconsistent?

Central schemes like JST uses artificial dissipation for the stabilization. This modification is an artificial one. Does this additional term makes system inconsistent? Can we expect this term to be ...
1
vote
1answer
57 views

How do I add some floating point numbers, keeping numerical accuracy in mind?

I am solving a problem involving the line with the set of points $(x_3,y_3)$ that are equidistant to two given points $(x_1,y_1)$ and $(x_2,y_2)$. The equation for this line is $$(x_3 - x_1)^2 + (y_3 ...
6
votes
1answer
108 views

How to derive an Implicit Runge-Kutta method from Pade approximation

I was reading some work by Butcher and I came across Pade approximations and the correlation between them and stability functions for some Implicit Runge-Kutta methods. For example, in this Pade table ...
2
votes
0answers
31 views

How to classify chaotic systems from a stability perspective

I am wondering what chaotic systems are from the perspective of numerical analysis. I am talking about 'deterministic chaos' such as for instance the 'logistic map' exhibits it. That is, the solution ...
3
votes
2answers
141 views

what do zero real parts of eigenvalues mean? Any good references?

I am solving a 1D advection problem of the the form $$dQ/dt=[A]Q$$ where {Q} is the vector of unknowns and [A] is the matrix of coefficients of spatial discretisation. I have worked out the ...
1
vote
0answers
39 views

comparison of stability of two non-linear methods

I have solved a numerical problem using two different sets of non-linear governing equations. I want to get an understanding of the stability of the methods relative to each other. To do so, I solving ...
2
votes
0answers
106 views

what do positive real parts of eigenvalues mean?

I am solving a 1D advection problem of the the form $$d{Q}/dt = [A]{Q}$$ where {Q} is the vector of unknowns and [A] is the matrix of coefficients of spatial discretisation. I have worked out the ...
2
votes
2answers
113 views

Why have specialised upwind schemes been developed to solve hyperbolic equations?

Are upwind schemes such as Godunov type methods superior to central differencing schemes? Do the reasons include superiority in modelling hyperbolic problems with Dirichlet BC's?
2
votes
0answers
73 views

Difference between fast and normal Givens Rotations?

would someone be so kind as to explain me the difference between the ordinary givens-rotation and the fast givens-rotation? I know that the fast givens Rotation reduces the Count of operations to ...
3
votes
1answer
71 views

linear stability analysis using spectral radius

I am analysing the stability of a series of 1D linear equations of the form \begin{equation} \frac{d}{dt} x = A x \end{equation} discretised using upwind and central finite volume methods, etc, with ...
3
votes
1answer
84 views

stabilizing advection-diffusion with multi-grid?

If one chooses to discetize the advection-diffusion (AD) equation using the standard Galerkin finite element method, stability issues may arise in cases of high Peclet number (i.e., high advection to ...
4
votes
0answers
64 views

What is the source of the error in the Sherman-Morrison formula application?

The Sherman-Morrison formula $$ (A+uv^T)^{-1} = A^{-1} - \frac{A^{-1}uv^TA^{-1}}{1+v^TA^{-1}u} $$ results in small errors in relation to the standard matrix inverse operation after each application, ...
1
vote
0answers
73 views

Reducing oscillations a 3D Alternating direction explicit scheme for the diffusion equation?

Hi I have made a 3D alternating direction explicit scheme for solving the diffusion equation, which will eventually replace a FTCS scheme in model of bubble dynamics in tissue. I have been testing it ...
7
votes
1answer
105 views

Forcing an ODE solver to preserve the norm

I have an ODE of the form $$ \frac{dy}{dt} = -i H y \enspace .$$ where $y$ is a complex vector and $H$ is a time dependent Hermitian matrix. The norm of the solution $y(t)$ at any point in time ...
3
votes
0answers
65 views

Stability analysis for explicit time discretization in the Finite Element Method

I have been looking for stability analysis of general reaction-diffusion problems, of the form $\frac{\partial u}{\partial t}=\nabla\cdot D\nabla u-k\,u$ , to be solved using the standard Finite ...
7
votes
1answer
110 views

Stability analysis of Heun's method

I am using Heun's method with a third order upwind spatial scheme, which is suggested by Shao (2008) to be used for solving the horizontal advection part of the advection-diffusion equation. This is ...
6
votes
0answers
57 views

Order of operations, numerical algorithms

I have read that (1) Ill conditioned operations should be performed before well conditioned ones. As an example, one should calculate $xz-yz$ as $(x-y)z$ since subtraction is ill conditioned ...
3
votes
1answer
187 views

Numerically stable approach for calculating x in Ax=b

I have an equation $Ax=b$ for which I need to solve for numerous $x$ matrices given $b$. Both $x$ and $b$ are nx1 matrices. Unfortunately, $A$ is a 32x32 matrix and inversion gives highly unstable ...
10
votes
1answer
184 views

Can an approximated Jacobian with finite differences cause instability in the Newton method?

I have implemented a backward-Euler solver in python 3 (using numpy). For my own convenience and as an exercise, I also wrote a small function that computes a finite difference approximation of the ...
0
votes
0answers
48 views

Backwards Stability

Let $A$ be a nonsingular matrix, let $x_{k+1}$ be an approximation to the solution of $Ax=b$, and let $r^{k+1}=b-Ax^{k+1}$. Show that if $x^{k+1}$ is $\epsilon$-backward stable approximate of ...
0
votes
4answers
339 views

Why are upwind schemes stable in convection flow calculation?

It is well known that upwind schemes are stable when calculating convection flows with $|\text{Pe}|>2$, $\text{Pe}$ is the Peclet number. Why is that, and why is central difference unstable? Is ...
1
vote
0answers
111 views

Tips on improving stability in numerical scheme for non-linear PDE

I am solving a non-linear second order system of PDEs in two variables. The equations are too complicated to write out here, but an essential feature is that there is a propagating wave which then ...
4
votes
3answers
131 views

Numerical solution of IVP for linear ODE with variable coefficient blows up

Cross posted in Mathematica.SE, I'll try to rephrase it in a more general way here. A friend of mine showed me this initial value problem (IVP) for a linear ordinary differential equation (ODE) with ...
0
votes
0answers
33 views

Locally evaluate nonlinear dynamic system's stability using eigenvalues

I'm working with Computational Neuroscience. I have a large Synaptic Matrix (x axis: presynaptic NeuronID, y axis: postsynaptic NeuronID) in a Modular network. This matrix is close to a random one and ...
6
votes
1answer
225 views

How to avoid negative values of numerical solution of transport equation using FEM scheme?

The transport equation is actually an advection-diffussion-reaction equation, which has the form as $$\frac{\partial C}{\partial t} + v_1 \frac{\partial C}{\partial x} + v_2 \frac{\partial ...
0
votes
1answer
99 views

Avoid arithmetic overflow in matrix multiplication

I am solving the following matrix equation for $\mathbf{x}$: $$(J^{\mathbf{T}}J)\mathbf{x}=J^{\mathbf{T}}\mathbf{r}$$ $J$ is $m\times n$ matrix $\mathbf{x}$ is vector of size $n$ $\mathbf{r}$ is ...
0
votes
0answers
88 views

Super time stepping and Crank Nicholson Method

I was wondering whether it is possible to combine the two to produce a very efficient code? Thanks,
3
votes
0answers
121 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 ...
1
vote
1answer
56 views

Adjusting Keplerian orbits for thrust with numerical stability

I'm writing a mod for a game that models orbital physics (Kerbal Space Program, or KSP). I'm attempting to model the effects of thrust on spacecraft in certain states where the game only models them ...
3
votes
2answers
91 views

How does constraint resolution affect the stability/accuracy of numerical integration?

I understand some basic analysis techniques (local truncation error, global error, zero-stable, absolute stable, etc.) of numerical integration. But I find it hard to apply these techniques in ...
14
votes
1answer
233 views

Puzzling remark about stability region of fifth-order Runge-Kutta method

I came across a puzzling remark in the paper P. J. van der Houwen, The development of Runge-Kutta methods for partial differential equations, Appl. Num. Math. 20:261, 1996 On lines 8ff on page 264, ...
3
votes
2answers
192 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 ...
7
votes
4answers
393 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
35 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 ...