# Questions tagged [time-integration]

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### Implicit integrator for ODE with quadratic right-hand side

I have an ODE for an unknown $x(t):[0,\infty)\to\mathbb R^n$ of the following form: $$x_i'(t)=a_i^\top x(t) + x(t)^\top Q_i x(t),$$ for $i\in\{1,\ldots,n\}$. Here, the vectors $a_i\in\mathbb R^n$ ...
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### Jos Stam's stable fluids — why is the timestep multiplied by the number of grid cells?

One more question about Jos Stam's GDC tutorial on stable fluids: in the advection step on page 8, the timestep for each dimension is implemented as dt * N, where <...
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### Conservation of energy test for 2-body problem

I'm trying to implement a C++ code for the evaluation of the solution of an N-body system of ODE. I've started with a 2-body problem just to set the methods ...
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### Second Order Time Integration for Stiff Linear System Avoiding any Explicit Step

I have a linear system $$\dot x(t) = Ax(t), \quad x(0)=x_0 \tag{*}$$ with $A$ being Hurwitz (i.e. the solutions may oscillate but will eventually tend to zero) but really stiff. $A$ might be large, ...
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### Parallel In Time with Multigrid

I am trying to solve the linear finite element equation $M\ddot{u}+Ku=F(t)$, where $M$ is the mass matrix ,$K$ the stiffness matrix and $F(t)$ the external load vector, parallel in time using XBraid ...
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### Numerical integration in time for finite elements

I am trying to solve $M\ddot{u}=-Ku+F_\text{ext}$ for a 2D linear elastic model with $M$ be the mass matrix,$K$ the stiffness matrix and $F_\text{ext}$ the external load vector coming from a uniformly ...
168 views

### How do I apply BDF2 in a STRANG splitting

I have a 3D diffusion equation that I want to solve using a time splitting (2D+1D). Assume that $A$ is the 2D discrete diffusion operator and $B$ is the 1D discrete diffusion operator. I want to use a ...
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### How to calculate the average velocity over a surface through time using paraview?

My problem is a little more complex, but here is a simplified similar model. Let us consider a fire source inside a box and the hot gases (burnt or unburnt) from the combustion are going out of the ...
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### Mass conservation for hyperbolic relaxation problem

I have solved numerically the following system: \begin{cases} \partial_t{u} + \partial_x{v} = 0 \\ \partial_t{v} + \frac{1}{\varepsilon^2}\partial_x{u} = -\frac{1}{\varepsilon^2}(v-f(u)) \end{cases} ...
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### The velocity Verlet method and variable time steps

Does the velocity Verlet handle variable time steps? I found controversial statements about it. In the paper Skeel, R. D., "Variable Step Size Destabilizes the Stömer/Leapfrog/Verlet Method", BIT ...
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### Integrating the 2d vorticity equation on periodic boundaries

This question is a follow-up of https://stackoverflow.com/questions/44718160/solve-a-linear-system-for-fft-coefficients At some time (kt), the FT of the vorticity (omega) satisfies: ...
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. ...