The tag has no wiki summary.

learn more… | top users | synonyms

9
votes
2answers
154 views

Is it well known that some optimization problems are equivalent to time-stepping?

Given a desired state $y_0$ and a regularization parameter $\beta \in \mathbb R$, consider the problem of finding a state $y$ and a control $u$ to minimize a functional \begin{equation} \frac{1}{2} ...
2
votes
1answer
63 views

Method of lines for inhomogeneous Dirichlet conditions

I understand how to set up the boundary conditions for a steady state problem discretized by Galerkin method, for a time dependent PDE below, $$\frac{\partial}{\partial t} u = c\nabla^2 u + a\nabla u ...
1
vote
1answer
74 views

Time Integration of a nonlinear reaction-diffusion system

I want to solve the following system of nonlinear reaction-diffusion equations (Schnakenberg Turing) using FEM methods (such as deal.ii): $$ \partial_{t} u = \Delta u + \gamma\left(a-u+u²v\right)$$ ...
7
votes
1answer
120 views

Algorithm to calculate the exponential of an Hessenberg matrix

I am interested in computing the solution of a lage system of ODEs using a krylov method as in [1]. Such method involve functions related to the exponential (the so-called $\varphi$-functions). It ...
2
votes
0answers
99 views

Integration of nonlinear PIDE via spectral methods

At the mean-field level, the dynamics of a polariton condensate can be described by a type of nonlinear Schrodinger equation (Gross-Pitaevskii-type), for a classical (complex-number) wavefunction ...
4
votes
1answer
245 views

Coupled nonlinear PDEs with time dependence on the RHS

I would like to numerically solve the following system of 2 coupled partial differential equations for the unknown functions $\psi_X(x,y,t)$ and $\psi_C(x,y,t)$: $\partial_t \psi_X = -i\psi_C - ...
0
votes
0answers
12 views

Model estimation using ACF and PACF

CAN ANYONE HELP IN MODEL ESTIMATION? The following are the ACF,PACF and the plot of the sample respectively.
3
votes
3answers
72 views

Numerical integration of non-uniform acceleration samples

I'm given a stream of acceleration data with timestamps. The sampling is non-uniform. Apart from Euler, is there a way to integrate the acceleration into velocity? Something more accurate or of ...
0
votes
1answer
131 views

Dual time stepping for fluid dynamics

I'm attempting to implement the Weiss and Smith preconditioner in an existing finite volume code and I am struggling with the idea of dual time stepping. My inner time steps are predictor-corrector, ...
1
vote
0answers
43 views

Stationary phase approximation for an integral with infinity saddle points?

I need a hand with the numerical evaluation, in Mathematica, for this integral: $$f(t)=\int_{-\infty}^\infty Exp\{it(\omega_H-\omega_l-\omega_k) - \sum _{j\neq(l,k)} S_j [1-e^{-it\omega_j}]\}\, dt$$ ...
1
vote
1answer
74 views

RATTLE numerical integrator example

I want to understand how the RATTLE algorithm works. Can somebody give me an example (in pseudocode or using any programming language like python or matlab) of how would I implement a numerical ...
6
votes
2answers
154 views

Spectral Methods in time

I was reading up on Spectral Methods for PDEs. In all the descriptions I read, while the position component is approximated via a Fourier series or other methods, the time component is still ...
5
votes
1answer
179 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
119 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 ...
6
votes
1answer
275 views

Why am I getting so much error for my Runge Kutta Fehlberg solver?

My current project is a reprogramming of a protein folding model involving the solution of thousands of ODEs in C++. I've been making some stop and start progress as I'm writing the solver to run ...
4
votes
2answers
172 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 ...
3
votes
0answers
100 views

Time-stepping for coupled nonlinear PDEs

What are good references for time-stepping of the coupled incompressible Navier-Stokes-heat equation (Boussinesq flow), $$ \begin{cases} \rho\left(\dot{\mathbf{u}} + \mathbf{u}\cdot\nabla ...
7
votes
1answer
175 views

What are the differences between Parareal, PITA, and PFASST?

The Parareal, PITA, and PFASST algorithms are all across-the-domain techniques for parallelizing the solution of time-dependent problems in time. What are the guiding principles behind these ...
7
votes
1answer
203 views

Navier-Stokes solver: How to adjust the time step based on non-linear terms?

My code solves the incompressible Navier-Stokes equation in a conducting fluid, together with the induction equation: $ \partial_t u + u \nabla u + 2\Omega \times u = -\nabla p + \nu \Delta u + ...
2
votes
0answers
57 views

Strong stability preserving RK scheme

For the ODE $$ \dot{x} = f(x) $$ we have the 2-stage, second order SSP RK scheme (Shu, Osher, Gottlieb) $$ x^{(0)} = x^n $$ $$ x^{(1)} = x^{(0)} + \Delta t f(x^{(0)}) $$ $$ x^{(2)} = \frac{1}{2} ...
4
votes
2answers
137 views

Improving the time integration of implicit discretized PDE with a non-linear source term

This might be a naive question, but when applying a implicit discretization to a PDE with a source term, should the source be averaged in time? For example if we take the diffusion equation with a ...
5
votes
1answer
91 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 ...
12
votes
2answers
904 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 ...
12
votes
1answer
240 views

What is the correct way of integrating in astronomy simulations?

I'm creating a simple astronomy simulator that should use Newtonian physics to simulate movement of planets in a system (or any objects, for that matter). All the bodies are circles in an Euclidean ...
2
votes
2answers
121 views

What are good examples of problems which are stiff due to very long interval of integration?

There is a class of stiff initial value problems for ODEs that have small Lipschitz constants, slowly-changing solutions, but very long interval of integration. The only practical example of such a ...
6
votes
1answer
121 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. ...
7
votes
1answer
150 views

Demonstrating that the time step size is small enough in a code with automatic step size selection

I recently inherited a large body of legacy code that solves a very stiff, transient problem. I would like to demonstrate that the spatial and temporal step sizes are small enough that the ...
10
votes
5answers
296 views

Examples of PDE computations using parallelism in both space and time

In the numerical solution of initial boundary value PDEs, it is very common to employ parallelism in space. It is much less common to employ some form of parallelism in the time discretization, and ...
6
votes
1answer
71 views

How do you change the desired accuracy of a TS object in PETSc?

I'm currently getting very long propagation times when attempting to use the Time Stepping propagators in Petsc 3.2, and in the interest of speeding things up, I'm curious how I can reduce the ...
9
votes
3answers
439 views

Can I use an explicit time stepping scheme to determine numerically whether an ODE is stiff?

I have an ODE: $u'=-1000u+sin(t)$ $u(0)=-\frac{1}{1000001}$ I know that this particular ODE is stiff, analytically. I also know that if we use an explicit (forward) time stepping method ...