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Questions tagged [time-integration]

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1
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1answer
76 views

Stability Criteria for Numerical Solution of Windkessel Ordinary Differential Equation

I'm trying to solve this equation (Windkessel equation) numerically as: $$C \frac{d P}{d t} + \frac{P}{R} = Q(t)$$ Where $C$ is compliance, $R$ is resistance, $P$ is pressure, and $Q(t)$ is a known ...
3
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0answers
43 views

Should I expect computational gains using a second-order splitting method here?

I am trying to solve a three-dimensional baroclinic transport problem. The hydrodynamic (three-dimensional shallow water) equations are: \begin{align} \nabla \cdot \vec{v} = 0, \tag{1} \\ \frac{\...
1
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1answer
50 views

Wanted: smoothing time domain transform

Let $A$ be a finite (and small-ish) set of positive real numbers and 0. Let $B$ be a subset of $\mathbb N^0$, up to some (small-ish) bound. I have a function $f(t)$, $A \rightarrow B$ that is ...
3
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2answers
84 views

Error control and sequence acceleration at the same time

In a posteriori error control for solving ODEs, one typically computes two different approximate solutions, one of which being "more accurate" and one of which being "less accurate". If $y_q^{n+1}$ is ...
4
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1answer
67 views

Symplectic Algorithms for Hamilton’s Equations as opposed to just Volume-Preserving

this might be a silly question, but if we’re trying to numerically solve Hamilton’s equations with some discrete scheme, sometimes when the scheme preserves phase space volume (Hamilton’s eqns are ...
1
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1answer
67 views

General questions regarding stability for time-integration of operator-split PDE systems

I am interested in solving ODE systems of the form \begin{align} \frac{\partial \vec{u}}{\partial t} = F(\vec{u}) \end{align} where $F$ is a nonlinear operator, $\vec{u}$ is a vector valued function ...
1
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1answer
49 views

How to isolate and test time discretization order of accuracy

I have a code that uses both spatial and time discretization/integration. For convergence analysis, I am wondering how one would test the order of accuracy of their ${time}$ integration scheme? I ...
1
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1answer
81 views

How to implement Galerkin Method of Lines / FEM with black box integrators in scipy

Suppose I have some time dependent PDE, which can be written in the strong form as $$ \frac{\partial u}{\partial t} + \mathcal{L}(u) = f $$ Where $\mathcal{L}$ is some differential operator. If I ...
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2answers
1k views

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 ...
0
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1answer
26 views

Confirmation of FSAL property for IMEX methods by Kennedy and Carpenter

This question is a continuation of Fourth order IMEX Runge-Kutta method and Implementation details for high order IMEX methods by Kennedy and Carpenter. I need confirmation that ARK3(2)4L[2]SA by ...
5
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2answers
195 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 ...
3
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2answers
253 views

Developping PDE with Python symbolically and numericaly

I feel like publishing some previous works from my PhD thesis. I was using Mathematica to build a system of 2N partial differential equations for 2N functions by symbolic spatial Taylor expansion, ...
0
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2answers
68 views

How do you apply boundary conditions in a time-stepping problem?

It looks to me like a very common problem, yet I haven't been able to find any practical guide on the subject despite many hours searching. Here is a clearer statement of my question: I have a ...
4
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0answers
83 views

Crank-Nicolson integrator: oscillations with complex matrix

I'm working on a Real-Time TDDFT implementation and I am currently comparing different propagation schemes for the propagation of the Kohn-Sham wave function, $$ \phi(t+\Delta t) = \hat{\mathcal{U}}\...
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0answers
39 views

Split-step-method for coupled equations

I have implemented a split-step-method for an equation of the shape $$\partial_z E = i\partial_x^2E+ic|E|^2E$$ resulting in a split into the linear part $$L=\partial_x^2$$ and the nonlinear part $$N=...
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0answers
82 views

Definition of CFL number in Arbitrary Lagrangian-Eulerian framework

In an Eulerian frame of reference, the CFL number is defined as $$\sigma=\frac{u \Delta t}{\Delta x}$$ with $u$ the magnitude of the fluid velocity. A restriction such as $\sigma<1$ for time ...
0
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1answer
30 views

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

I am trying to solve a system of non-linear index-1 DAEs in which the derivatives of the state variables, $x(t)$ are corrupted by additive noise, $w(t)$ (whose co-variance matrix is known). $\dot x(t)...
0
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2answers
107 views

implicit odes solution using fdm

I am solving a non linear second order implicit initial value problem using finite difference method, but my results do not converge. Please guide me with an example, how we can apply finite ...
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0answers
142 views

Double Integrating acceleration data to obtain position: 2 Problems

I have a data sample from an accelerometer from my phone (pretty bad accelerometer though). I'm trying to double integrate it in order to obtain the position as a function of time. I'm using a program ...
2
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2answers
490 views

Schrödinger equation with time dependent Hamiltonian

I need to solve the Schrödinger equation with a time dependent Hamiltonian $$i\hbar \frac{\partial}{\partial t} \Psi = \left[-\frac{\hbar^2}{2m}\nabla^2 +\frac{1}{2} k(t)(x^2+y^2) + V(r)\right]\Psi $...
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0answers
45 views

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: ...
15
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1answer
1k views

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. ...
1
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1answer
129 views

Fully discrete finite element method for 1D dynamic euler-bernoulli beam problem

I am trying to solve a 1D initial boundary value problem in MATLAB using Finite Elements with time stepping, for the purpose of learning scientific computing and to build up to more difficult problems....
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0answers
50 views

FEM/FVM/FD for structural modeling and stability issues due to large structural constants?

I've read that in modeling structures problems, the finite element method (FEM) is typically used. I am unfamiliar with FEM, but I am wondering, in particular, if using FEM, as opposed to finite ...
1
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1answer
194 views

Higher-order Verlet integration

I'm using a simple version of Verlet integration for a particle–particle interaction system with collisions. At the end of each iteration, I integrate like this: ...
5
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2answers
398 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 ...
0
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1answer
106 views

Role of non-hermitian coefficient matrices in the discretization of self-adjoint operators

What is the role of non-symmetric coefficient matrices in the solution of partial differential equations with self-adjoint operators? Particularly, I'm thinking about time-propagation of a linear ...
4
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1answer
386 views

Boundary condtions on nonlinear FEM time integration

I'm using the finite element method to obtain the time response of a structure harmonically excited. I'm using a linear displacement function to obtain the stiffness matrix and the consistent mass ...
12
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1answer
526 views

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 ...
-2
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1answer
206 views

Solving a differential equation using numerical methods and matlab

(a) Consider the following differential Equation $$Y'(t)=\frac{1}{1+t^{2}}-2[Y(t)]^2$$ $$Y(0)=0$$ The exact solution is $$Y(t)=\frac{t}{1+t^2}$$ Using the Euler method to solve the following ...
2
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1answer
79 views

High order time splitting methods

There are lots of higher order time splitting method as shown by the list with real and complex coefficients $a_i, b_i, c_i$: $$ [e^{c_s \Delta t \hat C}] e^{b_s \Delta t \hat B} e^{a_s \Delta t \...
1
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2answers
551 views

Strang splitting

I have recently come across the Strang splitting and have some questions. For the differential equation of the form $$ dy/dt = (L_1 + L_2)y$$ Strang splitting implement the time splitting as $$ \...
1
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0answers
67 views

Time integration for elastodynamics

I'd like to solve the elastodynamics equation for a problem with fast scales from an impact (although no shocks). $\rho\ddot{\mathbf{u}} - \nabla \cdot \sigma(\mathbf{u}) = \mathbf{f}$ In structural ...
3
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1answer
220 views

Numerical Methods for Solving a Fully Nonlinear Time-Dependent PDE?

Are there numerical methods of solving the following fully nonlinear time-dependent PDE: $$\nabla^2u\left(\textbf{r}(t), \dot{r}(t), t\right)=f\left(\textbf{r}(t), \dot{r}(t), t\right),$$ for $\textbf{...
2
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1answer
3k 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 ...
4
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1answer
167 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
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1answer
111 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" ...
3
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1answer
267 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 ...
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0answers
42 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
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0answers
119 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
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3answers
837 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 ...
6
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1answer
319 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
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1answer
126 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
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1answer
397 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
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0answers
152 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 ...
1
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0answers
87 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 ...
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0answers
382 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
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1answer
327 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
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1answer
214 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 ...