# Questions tagged [time-integration]

For questions about the particulars of solving differential equations with time as the independent variable.

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### Estimating the spectral radius when applying the method of lines

Some time integrators, notably the Runge-Kutta-Chebyshev method, implemented in the RKC code from Sommeijer & Verwer, gives the user an option to provide a callback with an estimate of the ...
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### Does this second-order implicit Runge-Kutta method have a name?

I am studying the time-integration of the following paper, Young, L. C. (1981). A finite-element method for reservoir simulation. Society of Petroleum Engineers Journal, 21(01), 115-128. A copy (PDF)...
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### How can I validate my time integration scheme for my dynamic linear elasticity FEM code with a manufactured solution or similar?

I am solving for a dynamic linear elasticity problem: \begin{equation} \dfrac{\partial^2 u}{\partial^2 t} - \nabla \cdot \sigma = f \end{equation} where $u \in R^2$ with sufficient BCs and ...
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### Why do I get an oscillatory solution when applying the implicit trapezoidal method to the linear diffusion equation?

I wish to solve the following equation, $$\frac{\partial f}{\partial t}=\frac{\partial}{\partial x}\left(D(x)\frac{\partial f}{\partial x}\right)$$ using an exponential integrator. I discretize this ...
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### Example: Velocity Verlet reduced accuracy

Velocity Verlet is often held to far more accurate than forward Euler while being no more expensive. Technically, this requires some degree of regularity on the potential. But, is there a convincing ...
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### Need help to fully understand SciPy's odeint's reported step sizes, eval times, # of funct calls & total proc. time (re. question in Astronomy SE)

A recent question in Astronomy SE Numerical Programming using odeint takes more than 17 minutes got me interested in looking closer at SciPy's odeint. The problem is a modified orbital mechanical ...
1 vote
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### Velocity Verlet leading to faster simulation than Euler in an n-Body simulation?

I have all the constants set to the same values for each set of code, G, the timestep, the masses of the planets etc. But using Velocity Verlet doesn't work unless I lower the gravitational constant ...
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### Why is velocity Verlet better than Verlet for gravity if it has a worse order of magnitude for the error term

Even though this method is more widely used than the simple Verlet method mentioned above, it unfortunately has an error term of O(Δt^2) , which is two orders of magnitude worse. That said, if you ...
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### Numerov Method with Time Varying Potential

Is it possible to use the Numerov method to solve the Time Dependent Schrodinger Equation ($\frac{i\partial\Psi(x, y, z, t)}{\partial t} = \nabla^2\Psi(x, y, z, t) + \Psi(x, y, z, t)V(x, y, z, t)$) ...
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### Best approach to solve this system of equations?

I have the following 1D (in space, that is) system of equations I would like to solve: \begin{equation} \rho_{fs}\frac{\partial x_{fs}}{\partial t} = h_m\left(W_a - W_{fs}\right) - D_{eff}\left(\...
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### Stabilized Many Stage Runge-Kutta methods instead of Local/Multirate Time Stepping

Locally refined meshes are often inevitable for accurate, yet feasible computations. In the context of time-dependent PDEs, however, this comes at the cost that (due to the CFL condition) reducing the ...
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### Passing additional arguments to odeint from torchdiffeq to solve an IVP

In Python I use the package torchdiffeq (as provided here) to solve initial value problems. Given an arbitrary function ...
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### Using velocity verlet algorithm for nbody simulation results in planet leaving orbit

I'm new to computational physics and attempting an n-Body simulator in java but for some reason the planet in orbit leaves the orbit when it reaches the fastest accelaration aka the closest point to ...
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### How does non-dimensionalization improve the behavior of ODE solvers?

I have a set of coupled ODEs that I'm solving numerically. The independent variable is time and runs from values of $10^{15}$ to $10^{17}$ in units of seconds. The state variables in their usual ...
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### Solving the time dependent Schrödinger equation with leapfrog integration in 1D

To my frustration I am struggling to implement leapfrog integration for the time dependent Schrödinger equation. To the best of my knowledge this was first explicitly done in "A fast explicit ...
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### Time Integrators for Water Wave Simulation

I am interested in using a numerical wave tank (NWT) to study the performance of various water wave dampers using MATLAB. I am looking at nonlinear water waves. My current NWT (2d, periodic) is based ...
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### FEM applied to heat equation and incompatible conditions

Consider the problem $$u_t - \Delta u = f \text{ on } \Omega\times (0,T)\\u=0 \text{ on } \partial \Omega\times (0,T) \\ u(x,0)=g(x) \text{ on } \Omega$$ with $g$ NOT vanishing on the boundary. If I ...
1 vote
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### Preserving conservation properties across time-integration schemes

I am interested in deriving an accurate, implicit discretization of a nonlinear advection-diffusion equation $$\partial_t f = \frac{1}{v^2} \partial_v J(f,v) \equiv F(f,v) \tag{\star}$$ with flux ...
286 views

### How to deal with solving coupled ODE systems where variables are updated multiple times within each timestep?

I'm solving a system of coupled ODEs using Euler integration for simplicity. To make this concrete, please see the (extremely simplified) minimal working example below in Python. Imagine we have a box ...
118 views

### Time & Space matlab discretization Finite Differences confusion

I have been trying to solve this equation and write the finite difference scheme in matlab for months, but I still am not successful. Given the KdV Equation $$\tag{1}u_{t} -6uu_x+u_{xxx}=0$$ I have ...
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### 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-...
1 vote
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### Numerical integrator for $a'(t)=e^{-a(t)}f(t)$

Suppose I know a function $f(t)$ and all its derivatives in $t$ in closed form. Given $a(0)$ and some $t_0>0$, I'm looking for an explicit integrator that can estimate $a(t_0)$, where $a(\cdot)$ ...
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### N-body correct scaling

I realized an usual way to scale an N-body problem for an N-body simulation is by choosing units such that gravitational constant $G = 1$, but I'm probably doing it the wrong way. Suppose I simply ...
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### How to select initial time step in adaptive time step ODE solver (TR-BDF2)

The Problem I am currently reconstructing a TR-BDF2 scheme which contains the following two stages: \begin{align} y_{n+\gamma} & = y_n + \gamma \frac{h}{2}\left( f_n + f_{n+\gamma} \right) \...