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3
votes
1answer
75 views

Comparison of velocity Verlet and leapfrog algorithms

Many sources present the Euler, Verlet, velocity Verlet, and leapfrog algorithms for integrating Newton's equations. Based on the order of accuracy, it is agreed that velocity Verlet, Verlet, and ...
3
votes
1answer
94 views

Solve an ODE with positivity-preserving property unconditionally

I have an ODE for a scalar function $u=u(t)$ of the form: $$ \frac{du}{dt}=L(u). $$ Here the function $L=L(u)$ satisfies: $$ L(0)=0, \quad L'(u)\le0. $$ Then it is easy to see that the solution $u=u(t)...
2
votes
1answer
51 views

How does Multi Body Dynamics software work for flexible joints?

I need to model a "fishing rod" in 2 dimensions by joining several "rigid sticks" by flexible/elastic joints. The joints act as plate/torsion springs with different spring constants. The "fishing rod" ...
4
votes
1answer
124 views

Computational time not proportional to integration interval in ODE-solver?

I am running octave and i have been trying ode45, ode54, ode23 etc to integrate the equation ` $$Q''(t) = B\cos(Q)\sin(\omega t)$$ $$Q(t=0)=0.$$ When the time interval to be integrated increases, the ...
4
votes
0answers
53 views

Lie Expansion Integration V/S Runge-Kutta Schemes

It is known that one can perform numerical integration of particular order by considering the Lie Expansion (http://adsabs.harvard.edu/full/1984A%26A...132..203H). For example \begin{equation} \frac{...
0
votes
0answers
18 views

Second and Higher Order Order Corrector in Spectral Deferred Correction

I am trying to work out a second order or higher order correction for the method of Spectral deferred Correction (SDC). Specifically using as a corrector a second order or third order multi-step. In ...
0
votes
0answers
45 views

What does a negative time stepping mean? (Adaptive time stepping)

Summary behind the problem: The following code aims at solving a static elasto-plastic problem. Like a 2D square mesh based on an elasto-plastic constitutive model like Von-Mises or Drucker-Prager ...
3
votes
1answer
134 views

Time step size for transient simulations of vortex shedding

I am simulating unsteady flow around a circular cylinder (using FLUENT). I am facing the following problem. Kindly please help me. OBJECTIVE: To find out the Reynolds number at which vortex shedding ...
4
votes
1answer
107 views

Fourth order IMEX Runge-Kutta method

I am looking for the Butcher tableau of a fourth order accurate Runge-Kutta method with IMEX splitting. I have been reading the ''classical'' paper on the subject by Ascher, Ruuth and Spiteri as well ...
3
votes
1answer
115 views

Why do planets move at the wrong speed in my solar system model?

Not 100% sure where this question belongs, since I'm not sure if the problem is code related or not. I've written a program to model the solar system. I am now testing its accuracy. I've checked ...
9
votes
1answer
160 views

Which numerical methods preserve time reversal symmetry?

If I have a physical system which contains a time reversal symmetry (for example a Hamiltonian $H(x,p)=p^2/2m + V(x)$ with $V(x)$ real) and I want to solve the differential equations which describe ...
2
votes
0answers
109 views

Is it worth switching to timesteppers provided by PETSc if I can't write down a Jacobian for my problem? Case study with “the amoeba” toy problem

I am considering using petsc4py instead of scipy.integrate.odeint (which is a wrapper for Fortran solvers) for a problem ...
0
votes
0answers
76 views

Why not solve molecular dynamics trajectories exactly?

The basis of Molecular Dynamics simulation is the integration of newton's equations of motion: $$\frac{F}{m} = \frac{d^2r}{dt^2}$$ I understand that there are various methods based on Taylor's ...
1
vote
0answers
97 views

Time discretization of wave equation

I am trying to model the seismic wave equation and have therefore been reading about discretization schemes and their stability. I recently came across an insightful paper on 'Galerkin FEM methods for ...
2
votes
1answer
91 views

BDF methods for implicit-explicit method

Are there BDF formulas like the ones given here but one that can be used with implicit-explicit discretization? The right hand side in those formulas is supposed to be implicitly discretized at the ...
4
votes
1answer
100 views

How to physically understand time dependent boundary conditions?

I am a beginner in Computational science and FEM. I came across some PDEs which implement time dependent boundary conditions. I am not able to visualize exactly a physical scenario of how that would ...
2
votes
2answers
80 views

PDEs appropriate for adaptive time stepping algorithms

I'm looking for some physical phenomena for which an adaptive time stepping algorithm would be ideal. A PDE or ODE that showed very large gradients in time at a small period of time and smoother ...
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 ...
6
votes
1answer
147 views

Newton iteration applied to nonlinear PDE

I'm having difficulty understanding how to apply Newton iteration to nonlinear PDEs and then use a fully implicit scheme to time step. For example, I want to solve Burgers equation $$u_{t} + u u_{x} -...
2
votes
1answer
64 views

Time integrator for a WENO scheme for advection equation

I'm implementing a finite-difference WENO scheme for a simple advection equation $$ \frac{\partial u}{\partial t} + \frac{\partial u}{\partial x} = 0 $$ with periodic boundary conditions. I'm not ...
1
vote
1answer
113 views

Quadrature order for finite elements and time dependent discontinuous Galerkin

When setting up a finite element system you have to use quadrature to calculate the integrals. I'm having trouble understanding what order rule to use. I know of some rules of thumb, for example with ...
6
votes
1answer
194 views

Test of 3rd-order vs 4th-order symplectic integrator with strange result

In my answer to a question on MSE regarding a 2D Hamiltonian physics simulation, I have suggested using a higher-order symplectic integrator. Then I thought it might be a good idea to demonstrate the ...
4
votes
2answers
79 views

Dissipative time-stepping scheme for first order in time system

When solving semi-discrete equations (originating from finite element models, for example), which are second-order in time of the form \begin{equation} M\ddot d + C\dot d + Kd = F, \end{equation} ...
3
votes
2answers
227 views

Numerical Solution of the Advection Dispersion equation

I am facing a simple (at first glance) problem. I need to implement a numerical scheme for the solution of the first order wave propagation equation with chromatic dispersion included. My original ...
6
votes
4answers
5k views

Why are Runge-Kutta and Euler's method so different?

I am solving a system of linear equations, $\underline {\dot x}=\underline A\cdot \underline x$, numerically. I have done this using the popular of methods of Euler and Runge-Kutta (RK). I have ...
1
vote
1answer
41 views

Resources exploring the problem of “volume exclusion”?

Consider the following situation: There are two boundaries -- one is denoted using grey lines, and the other is denoted using black lines. The boundaries are numerically represented using "vertices"...
0
votes
2answers
107 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
64 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 ...
5
votes
1answer
3k 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
131 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
339 views
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 ...
2
votes
1answer
53 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 ...
8
votes
2answers
227 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
124 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_{-1/2}^{1/2} \frac{\pi\cos u}{\sin\pi u-\sin\pi x} \frac{\...
1
vote
1answer
146 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
56 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
59 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 \partial\Omega\times(0,...
3
votes
2answers
230 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
2answers
231 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
517 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
36 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
253 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
193 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
1k 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
190 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) \; \; \dot{...
3
votes
2answers
92 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
223 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
235 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 ...
5
votes
2answers
250 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 ...