6 votes
Accepted

Step-size selection for an Trapezoidal Method ODE solver (ode23t)

The only documentation I know about for the implementation of ode23t is in the paper which documents the implementation of ode23tb, the TRBDF2 method in MATLAB. As ...
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5 votes
Accepted

Adaptive Timestepping for Stong Stability Preserving (SSP) Runge-Kutta Methods

SSP methods are mainly used for integrating ODEs corresponding to nonlinear hyperbolic PDE semi-discretizations. In such ODEs, $\Delta t_{FE}$ depends on the solution and so it varies at each time ...
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5 votes

Time integration of wave equation

It's not really meaningful to talk about integrating the equation in form A or B, since one way to integrate A is to first transform to B and then discretize. You can only really compare the actual ...
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5 votes
Accepted

Step size updating scheme adaptive embedded RK methods

First of all I don't see how this estimates the local error defined as the error we make in a single step using correct previous values. What does a lower order method have anything to do with ...
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4 votes
Accepted

Dormand–Prince 5(4): How to update the stepsize and make accept/reject decision?

The basic idea is that You use the estimated error given to you (cheaply) by the embedded methods; You use a metric to define acceptance using a user-defined relative and absolute tolerance; Based on ...
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  • 6,066
4 votes
Accepted

How to select initial time step in adaptive time step ODE solver (TR-BDF2)

Many numerical tips and theoretical explanations can be found in this book from Hairer and Wanner: https://www.springer.com/gp/book/9783540566700 In this book, a strategy is described, which uses a ...
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  • 1,085
4 votes

Automatic timestep adjustment in a CFD solver

For flow solvers, the general rule is that the time step needs to satisfy some kind of "CFL condition", named after Courant, Friedrichs, and Lewy. This means that $$ \Delta t \le C \min_{K} \frac{...
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3 votes

Automatic timestep adjustment in a CFD solver

Yes! Normally what's done is called Method of Lines. Essentially, you discretize in space to get all of your operators, but instead of discretizing the time component, you leave that derivative along. ...
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3 votes
Accepted

Open source solver for continuous-time stochastic non-linear DAEs (SDAEs)

DifferentialEquations.jl in Julia can do it if you can write it in mass-matrix form. You won't find it mentioned in the tutorial, but you can provide a mass matrix as part of the ...
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3 votes

Error control and sequence acceleration at the same time

What you're talking about is local extrapolation. Local extrapolation is the idea of getting an error estimate between two methods and then continuing (accepting the new value) from the higher order ...
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3 votes

Error control and sequence acceleration at the same time

You should state clearly what you mean by sequence acceleration. But if I understand you correctly, what you're asking about is exactly what extrapolation codes do. A sequence of low-order ...
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2 votes
Accepted

Is the time step size of a Rosenbrock method for stiff systems iteratively calculated?

Rosenbrock methods utilize embedded lower order methods in order to calculate errors for adaptive time stepping. In addition, Rosenbrock methods do not have to solve an implicit system (just a linear ...
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2 votes
Accepted

Testing Wiener process splitting in adaptive-step SDE integrators

I discuss the method you describe in more detail in this paper (Rackauckas and Nie 2017) as RSwM2. In that paper I am ever so slightly able to detect that it's sometimes doing something wrong, but ...
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1 vote
Accepted

ODE adaptive time stepping: is it bad to use "timescales of change" to select timestep size

As stated there's no re-rejection mechanism, i.e. ability to decrease the stepsize after a step has potentially failed. This is required for implicit methods which have Newton steps since there's a ...
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1 vote

ODE adaptive time stepping: is it bad to use "timescales of change" to select timestep size

I do not think this is a bad approach, but it is not a very precise way to select timesteps either. Admittedly, I have not come across this sort of timestep heuristic before, but looking at a linear ...
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1 vote
Accepted

Wanted: smoothing time domain transform

Look up monotone splines, e.g. Wikipedia Monotone_cubic_interpolation. (Normal cubic splines, see e.g. Numerical Recipes pages 120-122, are simpler but, surprisingly, may be non-monotone for monotone ...
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  • 892

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