Questions tagged [time-integration]
For questions about the particulars of solving differential equations with time as the independent variable.
147
questions
2
votes
0
answers
91
views
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 ...
- 21
1
vote
0
answers
58
views
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
...
- 695
0
votes
1
answer
78
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 ...
- 111
0
votes
1
answer
68
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 ...
- 23
1
vote
0
answers
113
views
Numerical evaluation of Duhamel's integration
I am trying to numerically evaluate the following Duhamel's integration:
$$
x = \frac{-1}{\omega_d} \int_0^t \ddot{x}_g (\tau) e^{-\zeta \omega_n(t - \tau)} \sin{\left( \omega_d (t - \tau) \right)} d\...
- 425
2
votes
1
answer
85
views
Implicit integrator for ODE with quadratic right-hand side
I have an ODE for an unknown $x(t):[0,\infty)\to\mathbb R^n$ of the following form:
$$
x_i'(t)=a_i^\top x(t) + x(t)^\top Q_i x(t),
$$
for $i\in\{1,\ldots,n\}$. Here, the vectors $a_i\in\mathbb R^n$ ...
- 1,331
3
votes
1
answer
57
views
Jos Stam's stable fluids — why is the timestep multiplied by the number of grid cells?
One more question about Jos Stam's GDC tutorial on stable fluids: in the advection step on page 8, the timestep for each dimension is implemented as dt * N, where <...
- 145
5
votes
1
answer
181
views
Write incompressible Navier Stokes as ODE in $(\mathbf{u},p)$
Consider the Navier stokes equation after the discretization with conforming finite elements with source term $f=0$. We have the algebraic structure of a saddle point problem:
$$M \dot{u} = f- Au -B^...
- 271
0
votes
0
answers
80
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^{-...
- 591
0
votes
0
answers
48
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(...
- 101
5
votes
0
answers
184
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-...
- 425
1
vote
1
answer
98
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 $a(\cdot)$ ...
- 1,331
2
votes
0
answers
49
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 ...
- 31
-1
votes
1
answer
139
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
$$\\\frac{\partial u}{\partial t} + C(x, t) \frac{\partial u}{\partial x} = 0 \\\\$$
$$ \\ C(x,t) = \...
- 1
0
votes
0
answers
146
views
Conservation of energy test for 2-body problem
I'm trying to implement a C++ code for the evaluation of the solution of an N-body system of ODE. I've started with a 2-body problem just to set the methods ...
- 31
2
votes
1
answer
129
views
Second Order Time Integration for Stiff Linear System Avoiding any Explicit Step
I have a linear system
$$
\dot x(t) = Ax(t), \quad x(0)=x_0 \tag{*}
$$
with $A$ being Hurwitz (i.e. the solutions may oscillate but will eventually tend to zero) but really stiff. $A$ might be large, ...
- 3,398
0
votes
1
answer
66
views
Handling time derivative as source term in SDIRK methods
I am currently facing some challenge implementing a less traditional PDE which take a form similar to the Navier-Stokes equation, except that the continuity equation is modified such that:
$$\epsilon \...
- 1,117
2
votes
1
answer
237
views
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) \...
- 142
2
votes
1
answer
104
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 ...
- 73
3
votes
0
answers
86
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)
$$
\Delta t \approx \frac{h}{N^2} \frac{C}{|u| + |c| + \frac
{N^2 \mu}{h}}
$$
($...
- 41
1
vote
0
answers
52
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 ...
- 471
2
votes
2
answers
192
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:
$$
k =
\begin{cases}
1 ~\text{if} ~x &...
- 591
2
votes
0
answers
151
views
Parallel In Time with Multigrid
I am trying to solve the linear finite element equation $M\ddot{u}+Ku=F(t)$, where $M$ is the mass matrix ,$K$ the stiffness matrix and $F(t)$ the external load vector, parallel in time using XBraid ...
- 471
3
votes
1
answer
87
views
Numerical integration in time for finite elements
I am trying to solve $M\ddot{u}=-Ku+F_\text{ext}$ for a 2D linear elastic model with $M$ be the mass matrix,$K$ the stiffness matrix and $F_\text{ext}$ the external load vector coming from a uniformly ...
- 471
2
votes
1
answer
184
views
How do I apply BDF2 in a STRANG splitting
I have a 3D diffusion equation that I want to solve using a time splitting (2D+1D). Assume that $A$ is the 2D discrete diffusion operator and $B$ is the 1D discrete diffusion operator.
I want to use a ...
- 53
0
votes
0
answers
272
views
How to calculate the average velocity over a surface through time using paraview?
My problem is a little more complex, but here is a simplified similar model.
Let us consider a fire source inside a box and the hot gases (burnt or unburnt) from the combustion are going out of the ...
- 1
3
votes
2
answers
158
views
Mass conservation for hyperbolic relaxation problem
I have solved numerically the following system:
\begin{cases}
\partial_t{u} + \partial_x{v} = 0 \\
\partial_t{v} + \frac{1}{\varepsilon^2}\partial_x{u} = -\frac{1}{\varepsilon^2}(v-f(u))
\end{cases}
...
- 133
3
votes
1
answer
728
views
The velocity Verlet method and variable time steps
Does the velocity Verlet handle variable time steps? I found controversial statements about it.
In the paper Skeel, R. D., "Variable Step Size Destabilizes the Stömer/Leapfrog/Verlet Method", BIT ...
- 133
2
votes
1
answer
177
views
Lumped matrices in thermal analysis using finite elements
The governing equation of the transient heat transfer problem is
$$C \frac{dT}{dt}+K T = Q$$
$C$ is the heat capacity matrix. $K$ is the thermal conductivity matrix. $T$ is the temperature vector. $...
- 784
0
votes
0
answers
97
views
Lax-wendroff for stiff source terms
I am interested in problems of the form
$$
u_t = F(u) + S(u)
$$
where $F(u) = - div(f(u))$ and $S(u)$ is a stiff source term. I am looking for any existing works which develop Lax-Wendroff type ...
- 2,983
1
vote
2
answers
64
views
Numerical integral with symbolic integral in exponent
Many times in fourier approximation we come across integrals such as
$$\int_0^1 e^{-\gamma\int_0^xu_0(\eta)d\eta}dx$$ where $\gamma$ is a constant and the data for $u_0$ is provided as a discretely ...
- 149
3
votes
1
answer
164
views
Time integration of wave equation
My question is: how come that certain formulations of the wave equation can be time integrated more efficiently then others?
Le me expand a bit on that. Consider the wave equation:
$$ \frac{d^2 p(t,...
- 382
0
votes
0
answers
37
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....
- 216
1
vote
1
answer
133
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 ...
- 173
3
votes
0
answers
72
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{\...
- 379
1
vote
1
answer
64
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 ...
- 435
3
votes
2
answers
98
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 ...
- 379
4
votes
1
answer
94
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 ...
- 143
1
vote
1
answer
105
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 ...
- 379
1
vote
1
answer
96
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 ...
- 321
1
vote
1
answer
104
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 ...
- 642
41
votes
2
answers
8k
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 ...
- 842
0
votes
1
answer
60
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 ...
- 119
5
votes
2
answers
624
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 ...
- 51
3
votes
2
answers
649
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, ...
- 163
0
votes
2
answers
152
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 ...
- 141
3
votes
0
answers
158
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,
$$
\phi(t+\Delta t) = \hat{\mathcal{U}}\...
- 83
1
vote
0
answers
97
views
Split-step-method for coupled equations
I have implemented a split-step-method for an equation of the shape
$$\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=...
- 533
1
vote
0
answers
223
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 ...
- 1,055
1
vote
1
answer
54
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 covariance matrix is known).
$\dot x(t) =...
