# Questions tagged [time-integration]

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### What does “symplectic” mean in reference to numerical integrators, and does SciPy's odeint use them?

In this comment I wrote: ...default SciPy integrator, which I'm assuming only uses symplectic methods. in which I am refering to SciPy's odeint, which uses ...
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### BDF vs implicit Runge Kutta time stepping

Are there any reasons for why one should choose high order implicit Runge Kutta (IMRK) over BDF time stepping? BDF seems much easier to me as $q$ stage IMRK needs $q$ linear solves per time step. ...
895 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} \...
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### 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 ...
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### 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. ...
839 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 ...
571 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 ...
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### Why is leapfrog integration symplectic and RK4 not, if the latter is more accurate?

In a system where energy theoretically should be conserved, the most accurate simulation would conserve energy (as well as giving accurate positions, velocities and etc). RK4 is more accurate than ...
931 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, ...
914 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 (...
2k 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 ...
797 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 methods?...
49k 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 ...
490 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 . Such method involve functions related to the exponential (the so-called $\varphi$-functions). It ...
978 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 ...
1k 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 ...
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### 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 ...
106 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 ...
558 views

### Do there exist low-storage Runge–Kutta methods with an order larger than four?

I’m trying to find a Runge–Kutta integrator that only requires the absolute minimum of storage, i.e. it fulfills one of the following three, presumably equivalent criteria: Each evaluation of the ...
225 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 non-...
602 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 ...
2k 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 ...
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### 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 ...
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### 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 ...
123 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} ...
593 views

### Time discretization: Runge-Kutta methods vs. standard backward difference

I've recently written a code that solves the incompressible/low-Mach number formulation of the Navier-Stokes equation with high-order methods for both time and space. My advisor insisted that I use ...
341 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 ...
76 views

### Stability of Crank-Nicholson for advection diffusion equation for spatial discretization other than finite differences second-order centered

Crank Nicholson is a time discretization method (see 4th equation here). From what I see around, you can use different space discretization, such as Finite elements. But for the linear advection-...