Questions tagged [time-integration]
The time-integration tag has no usage guidance.
136
questions
0
votes
0answers
72 views
Transient advection equation with stabilized FEM
I am interested in solving the transient advection equation
$\left\{\begin{array}{ll}\partial_{t} u+\beta \cdot \nabla u=f & \text { in } \Omega, t>0 \\ u=0 & \text { on } \partial \Omega^{-...
0
votes
0answers
34 views
Expression for numerical amplification factor for Euler time integration and CD2 scheme for one degree hyperbolic function
This is the expression to be derived but I am not getting the exact expression as given in the image. Is the given expression given for the implicit Euler method?
Here $N_c = c (\delta t)/h$ and $\tan(...
4
votes
0answers
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-...
1
vote
1answer
85 views
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 $a(\cdot)$ ...
1
vote
0answers
40 views
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 ...
-1
votes
1answer
77 views
Numerical solution of the advection equation with Crank–Nicolson finite difference method
I need to implement a numerical scheme for the solution of the one-dimensional advection equation
$$\\\frac{\partial u}{\partial t} + C(x, t) \frac{\partial u}{\partial x} = 0 \\\\$$
$$ \\ C(x,t) = \...
0
votes
0answers
123 views
Conservation of energy test for 2-body problem
I'm trying to implement a C++ code for the evaluation of the solution of an N-body system of ODE. I've started with a 2-body problem just to set the methods ...
1
vote
1answer
110 views
Second Order Time Integration for Stiff Linear System Avoiding any Explicit Step
I have a linear system
$$
\dot x(t) = Ax(t), \quad x(0)=x_0 \tag{*}
$$
with $A$ being Hurwitz (i.e. the solutions may oscillate but will eventually tend to zero) but really stiff. $A$ might be large, ...
0
votes
1answer
63 views
Handling time derivative as source term in SDIRK methods
I am currently facing some challenge implementing a less traditional PDE which take a form similar to the Navier-Stokes equation, except that the continuity equation is modified such that:
$$\epsilon \...
2
votes
1answer
93 views
Best practice for ADTs in computational science with Fortran
I have been writing a software package in Fortran for solution of the Vlasov-Poisson system in 2D2V. I want this software to be useful beyond its current application (e.g. systems with different ...
3
votes
0answers
70 views
Explicit DG time step restriction for compressible Navier-Stokes equations
Hesthaven's book 1 mentions the following time step restriction for Navier-Stokes equations (see (7.32) in 2008 edition)
$$
\Delta t \approx \frac{h}{N^2} \frac{C}{|u| + |c| + \frac
{N^2 \mu}{h}}
$$
($...
1
vote
0answers
51 views
Multigrid Reduction In Time Convergence
I am trying to solve a 2D dynamic linear elasticity model parallel in time using Xbraid. The spatial domain is [0,1]x[0,1] and time domain [0,1]. For time integration I am using a backward Euler ...
2
votes
2answers
186 views
Heat equation in non-dimensional form behaving differently than in usual format
Starting from
$$
c_p \frac{\partial u }{\partial t} = k \nabla^2 u
$$
in a one dimensional domain [0,1] where $c_p$ and $k$ are modeling two different materials:
$$
k =
\begin{cases}
1 ~\text{if} ~x &...
2
votes
0answers
151 views
Parallel In Time with Multigrid
I am trying to solve the linear finite element equation $M\ddot{u}+Ku=F(t)$, where $M$ is the mass matrix ,$K$ the stiffness matrix and $F(t)$ the external load vector, parallel in time using XBraid ...
3
votes
1answer
83 views
Numerical integration in time for finite elements
I am trying to solve $M\ddot{u}=-Ku+F_\text{ext}$ for a 2D linear elastic model with $M$ be the mass matrix,$K$ the stiffness matrix and $F_\text{ext}$ the external load vector coming from a uniformly ...
1
vote
1answer
137 views
How do I apply BDF2 in a STRANG splitting
I have a 3D diffusion equation that I want to solve using a time splitting (2D+1D). Assume that $A$ is the 2D discrete diffusion operator and $B$ is the 1D discrete diffusion operator.
I want to use a ...
0
votes
0answers
159 views
How to calculate the average velocity over a surface through time using paraview?
My problem is a little more complex, but here is a simplified similar model.
Let us consider a fire source inside a box and the hot gases (burnt or unburnt) from the combustion are going out of the ...
3
votes
2answers
151 views
Mass conservation for hyperbolic relaxation problem
I have solved numerically the following system:
\begin{cases}
\partial_t{u} + \partial_x{v} = 0 \\
\partial_t{v} + \frac{1}{\varepsilon^2}\partial_x{u} = -\frac{1}{\varepsilon^2}(v-f(u))
\end{cases}
...
3
votes
1answer
512 views
The velocity Verlet method and variable time steps
Does the velocity Verlet handle variable time steps? I found controversial statements about it.
In the paper Skeel, R. D., "Variable Step Size Destabilizes the Stömer/Leapfrog/Verlet Method", BIT ...
2
votes
1answer
129 views
Lumped matrices in thermal analysis using finite elements
The governing equation of the transient heat transfer problem is
$$C \frac{dT}{dt}+K T = Q$$
$C$ is the heat capacity matrix. $K$ is the thermal conductivity matrix. $T$ is the temperature vector. $...
0
votes
0answers
81 views
Lax-wendroff for stiff source terms
I am interested in problems of the form
$$
u_t = F(u) + S(u)
$$
where $F(u) = - div(f(u))$ and $S(u)$ is a stiff source term. I am looking for any existing works which develop Lax-Wendroff type ...
1
vote
2answers
60 views
Numerical integral with symbolic integral in exponent
Many times in fourier approximation we come across integrals such as
$$\int_0^1 e^{-\gamma\int_0^xu_0(\eta)d\eta}dx$$ where $\gamma$ is a constant and the data for $u_0$ is provided as a discretely ...
3
votes
1answer
142 views
Time integration of wave equation
My question is: how come that certain formulations of the wave equation can be time integrated more efficiently then others?
Le me expand a bit on that. Consider the wave equation:
$$ \frac{d^2 p(t,...
0
votes
0answers
33 views
What will PDE discretization matrix look like for time and space? [duplicate]
Please note: this question is not a duplicate of this question since, while the PDE is the same, the nature of this question is different, i.e. the other question treats a different aspect of this PDE....
1
vote
1answer
119 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
votes
0answers
67 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
vote
1answer
63 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
votes
2answers
95 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
votes
1answer
86 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
vote
1answer
96 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
vote
1answer
79 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
vote
1answer
98 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 ...
34
votes
2answers
6k 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
votes
1answer
53 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
votes
2answers
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 ...
3
votes
2answers
542 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
votes
2answers
132 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 ...
3
votes
0answers
156 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,
$$
\phi(t+\Delta t) = \hat{\mathcal{U}}\...
1
vote
0answers
83 views
Split-step-method for coupled equations
I have implemented a split-step-method for an equation of the shape
$$\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=...
1
vote
0answers
170 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
votes
1answer
41 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
votes
2answers
154 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 ...
1
vote
0answers
274 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
votes
2answers
727 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 $...
1
vote
0answers
58 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:
...
20
votes
1answer
3k 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
vote
1answer
218 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....
1
vote
0answers
65 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
vote
1answer
361 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:
...
1
vote
0answers
313 views
