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-1
votes
2answers
78 views

Looking for a particular algorithm for numerical integration

Consider the following differential equation \begin{equation} p(t) = \frac{\partial q(t)}{\partial t} \end{equation} where $t \in (0,\infty)$. I have a build a code that spits out values of the ...
0
votes
0answers
44 views

How to keep velocities in check in molecular dynamics simulation?

I am trying to make a very simple molecular dynamics simulator with Reflective Boundary Conditions. I am assigning the initial positions in a cube randomly while making sure they are not too close to ...
4
votes
1answer
129 views

How to formulate lumped mass matrix in FEM

When solving time dependent PDE's using the finite element method, for example say the heat equation, if we use explicit time stepping then we have to solve a linear system because of the mass matrix. ...
1
vote
0answers
38 views

scipy.integrate.ode ignores boundary conditions

I am trying to solve the 1-dimensional diffusion problem numerically using method of lines: $$ \frac{\partial c}{\partial t} =D \frac{\partial^2 c}{\partial z^2},$$ where the right hand side is ...
0
votes
1answer
57 views
4
votes
3answers
110 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 ...
2
votes
1answer
44 views

Numerical integration of a function whose expression is unknown

I want to compute the value of an integral of a function. This function, however, is not given by a formula, say $f(x) \: \forall x \in [0,1]$, but is only known through its values on some given ...
6
votes
1answer
79 views

Space-time finite element discretization for time-dependent PDEs

In FEM literature, semi-variational methods are typically used in the solution of time-dependent PDEs. I have not seen a fully-variational approach i.e. where space and time are discretised by FEM, ...
4
votes
0answers
89 views

numerical analysis of a partial integro-differential equation

I have to numerically solve a nonlinear partial integro-differential equation. This is my equation, $$\frac{\partial y(x,t)}{\partial t}=\int_{-\infty}^\infty K_0(|x-u|) \frac{\partial^2 ...
1
vote
1answer
141 views

Step-wise finite element formulations: can this be done?

Given the functional: $$ F[\mathbf{x}]=\frac{1}{2}[\mathbf{x}^{\text{T}} * D(\mathbf{x})]-\frac{1}{2}[\mathbf{x}^{\text{T}} * \mathbf{Ax}]-\frac{1}{2}\mathbf{x}^{\text{T}}(0)\mathbf{x}(t) $$ Where ...
1
vote
0answers
48 views

approximation of nonlinear time-dependent system with history

I have two time-dependent coupled equations. One of which is several orders of magnitude more computationally demanding than the other. I am trying to use machine learning to reproduce the behavior of ...
0
votes
1answer
56 views

Solving Initial Value problem ignoring the time-derivative

I am looking at a heat initial value problem \begin{align} \frac{\partial u}{\partial t}-\nabla^2u = f\quad&\text{in}\quad \Omega\times(0,T)\\ u = g \quad&\text{on}\quad ...
3
votes
2answers
145 views

Initial Value Problem using Finite Element

I am trying to implement a FEM solver for the following initial value problem \begin{align} \frac{\partial u}{\partial t} - \nabla^2 u &= f\quad \text{ in } \Omega\times (0,T)\\ u &= g\quad ...
0
votes
0answers
77 views

Calculating Conditional Expectations

I would like to compute the conditional expectation of a particular function : $E_{t}\left[ R(A_{t+1}, \eta_{t+1})\right | A_t]$ where $A_{t+1}$ and $\eta_{t+1}$ are two rv correlated to each ...
0
votes
2answers
94 views

Structural FEM analysis: transiet response vs frequency response

I am running 2 simulations on a cantilever plate in Nastran: one is a transient analysis (time domain) and the other one is a frequency response analysis. The transient analysis computes the response ...
1
vote
0answers
214 views

How to solve an ode with stochastic time-dependent input

I am trying to repeat an example I found in a paper. I have to solve this ODE: $25 \ddot{x} + 15 \dot{x} + 330000 x = p(t)$ where $p(t)$ is a white noise sequence band-limited into the 10-25 Hz ...
1
vote
0answers
31 views

Eigenvalues and Timestep restriction Follow up

This is a follow-up question to the previous questions I had on eigenvalues. Please let me know if I should edit the previous question itself for asking this. If the eigenvalues of a matrix ...
3
votes
1answer
173 views

Eigenvalues and Timestep restriction

For an equation of the kind $u_t=Au$, time step is usually defined by ensuring that the eigenvalues are within the stability region of the time-stepping scheme employed. If the eigenvalues are on ...
1
vote
1answer
185 views

How to write a function-generating code in Python/MATLAB?

I want to write a code that generates a function I(t) that satisfies the following condition: $\frac{\big<I(t)^2\big>}{\big<I(t)\big>^2} > 2$ In other words, $\frac{\lim_{T \to ...
3
votes
0answers
594 views

Numerical integration and filtering of acceleration experimental data

I have a vector containing acceleration measurements and the corresponding vector of times in which measurements are taken. To obtain velocity and displacement I used the cumtrapz() function already ...
6
votes
2answers
137 views

How does the L-stability or A-stability of a scheme relate to its ability to preserve a quadratic invariant?

I am working with the simple example of an oscillator: $$(1) \; \; \ddot{u} + u = 0, \; \; u(0) = u_0$$ I know that Forward Euler does not preserve an invariant of the above system: $$(2) \; \; ...
3
votes
2answers
82 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 ...
4
votes
1answer
194 views

Behavior of integration method

I was playing with N-body simulations of a game called Kerbal Space Program, which itself uses the patched conics approximation. I have read that for long term stability it is best to use symplectic ...
4
votes
1answer
214 views

Non-conservative implementation implicit Euler

In Matlab R2013a I have implemented the Implicit Euler (time) integration scheme. To find the $x^{n+1}$ value I use fixed point iterations: $x^{n+1} = \Delta t f(x^{n+1}) + x^n$ To test this, I use ...
4
votes
3answers
161 views

Machine precision and local error

I'm working with an RKF45 integrator that I have programmed using CUDA C++ on my GPU and am pondering a few questions as I'm trying to track down some issues with my code. I'm using double ...
11
votes
3answers
299 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
151 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 ...
3
votes
1answer
132 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
208 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
125 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
287 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
15 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.
4
votes
4answers
247 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
365 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
66 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
120 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
184 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
396 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
156 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
527 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
411 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
121 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
321 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
255 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
79 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
172 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
129 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 ...
13
votes
2answers
2k 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
369 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
132 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 ...