Questions tagged [time-integration]
The time-integration tag has no usage guidance.
127
questions
2
votes
1answer
68 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 ...
2
votes
0answers
55 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
34 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
158 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:
$$
k =
\begin{cases}
1 ~\text{if} ~x &...
2
votes
0answers
148 views
Parallel In Time with Multigrid
I am trying to solve the linear finite element equation $M\ddot{u}+Ku=F(t)$, where $M$ is the mass matrix ,$K$ the stiffness matrix and $F(t)$ the external load vector, parallel in time using XBraid ...
3
votes
1answer
77 views
Numerical integration in time for finite elements
I am trying to solve $M\ddot{u}=-Ku+F_\text{ext}$ for a 2D linear elastic model with $M$ be the mass matrix,$K$ the stiffness matrix and $F_\text{ext}$ the external load vector coming from a uniformly ...
1
vote
0answers
56 views
How do I apply BDF2 in a STRANG splitting
I have a 3D diffusion equation that I want to solve using a time splitting (2D+1D). Assume that $A$ is the 2D discrete diffusion operator and $B$ is the 1D discrete diffusion operator.
I want to use a ...
0
votes
0answers
55 views
How to calculate the average velocity over a surface through time using paraview?
My problem is a little more complex, but here is a simplified similar model.
Let us consider a fire source inside a box and the hot gases (burnt or unburnt) from the combustion are going out of the ...
3
votes
2answers
143 views
Mass conservation for hyperbolic relaxation problem
I have solved numerically the following system:
\begin{cases}
\partial_t{u} + \partial_x{v} = 0 \\
\partial_t{v} + \frac{1}{\varepsilon^2}\partial_x{u} = -\frac{1}{\varepsilon^2}(v-f(u))
\end{cases}
...
3
votes
1answer
273 views
The velocity Verlet method and variable time steps
Does the velocity Verlet handle variable time steps? I found controversial statements about it.
In the paper Skeel, R. D., "Variable Step Size Destabilizes the Stömer/Leapfrog/Verlet Method", BIT ...
2
votes
1answer
96 views
Lumped matrices in thermal analysis using finite elements
The governing equation of the transient heat transfer problem is
$$C \frac{dT}{dt}+K T = Q$$
$C$ is the heat capacity matrix. $K$ is the thermal conductivity matrix. $T$ is the temperature vector. $...
0
votes
0answers
74 views
Lax-wendroff for stiff source terms
I am interested in problems of the form
$$
u_t = F(u) + S(u)
$$
where $F(u) = - div(f(u))$ and $S(u)$ is a stiff source term. I am looking for any existing works which develop Lax-Wendroff type ...
1
vote
2answers
53 views
Numerical integral with symbolic integral in exponent
Many times in fourier approximation we come across integrals such as
$$\int_0^1 e^{-\gamma\int_0^xu_0(\eta)d\eta}dx$$ where $\gamma$ is a constant and the data for $u_0$ is provided as a discretely ...
3
votes
1answer
122 views
Time integration of wave equation
My question is: how come that certain formulations of the wave equation can be time integrated more efficiently then others?
Le me expand a bit on that. Consider the wave equation:
$$ \frac{d^2 p(t,...
0
votes
0answers
32 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
104 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
66 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
56 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
94 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
80 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
88 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
66 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
95 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 ...
32
votes
2answers
4k 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
43 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
451 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
443 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
103 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
146 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,
$$
\phi(t+\Delta t) = \hat{\mathcal{U}}\...
1
vote
0answers
72 views
Split-step-method for coupled equations
I have implemented a split-step-method for an equation of the shape
$$\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
145 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
145 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
262 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
684 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
52 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:
...
19
votes
1answer
2k 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
184 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
62 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
307 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
281 views
4
votes
2answers
550 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
votes
1answer
130 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 ...
5
votes
2answers
475 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 ...
13
votes
1answer
1k 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
votes
1answer
241 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}$$
Using the Euler method to solve the following ...
2
votes
1answer
88 views
High order time splitting methods
There are lots of higher order time splitting method as shown by the list with real and complex coefficients $a_i, b_i, c_i$:
$$
[e^{c_s \Delta t \hat C}] e^{b_s \Delta t \hat B} e^{a_s \Delta t \...
1
vote
2answers
1k 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
$$
\...
1
vote
0answers
73 views
Time integration for elastodynamics
I'd like to solve the elastodynamics equation for a problem with fast scales from an impact (although no shocks).
$\rho\ddot{\mathbf{u}} - \nabla \cdot \sigma(\mathbf{u}) = \mathbf{f}$
In structural ...
2
votes
1answer
261 views
Numerical Methods for Solving a Fully Nonlinear Time-Dependent PDE?
Are there numerical methods of solving the following fully nonlinear time-dependent PDE:
$$\nabla^2u\left(\textbf{r}(t), \dot{r}(t), t\right)=f\left(\textbf{r}(t), \dot{r}(t), t\right),$$
for $\textbf{...
