Questions tagged [time-integration]
For questions about the particulars of solving differential equations with time as the independent variable.
163
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Estimating the spectral radius when applying the method of lines
Some time integrators, notably the Runge-Kutta-Chebyshev method, implemented in the RKC code from Sommeijer & Verwer, gives the user an option to provide a callback with an estimate of the ...
- 454
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Does this second-order implicit Runge-Kutta method have a name?
I am studying the time-integration of the following paper,
Young, L. C. (1981). A finite-element method for reservoir simulation. Society of Petroleum Engineers Journal, 21(01), 115-128.
A copy (PDF)...
- 454
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0
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How can I validate my time integration scheme for my dynamic linear elasticity FEM code with a manufactured solution or similar?
I am solving for a dynamic linear elasticity problem:
\begin{equation}
\dfrac{\partial^2 u}{\partial^2 t} - \nabla \cdot \sigma = f
\end{equation}
where $ u \in R^2 $ with sufficient BCs and ...
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Why do I get an oscillatory solution when applying the implicit trapezoidal method to the linear diffusion equation?
I wish to solve the following equation,
$$\frac{\partial f}{\partial t}=\frac{\partial}{\partial x}\left(D(x)\frac{\partial f}{\partial x}\right)$$
using an exponential integrator.
I discretize this ...
- 87
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Example: Velocity Verlet reduced accuracy
Velocity Verlet is often held to far more accurate than forward Euler while being no more expensive. Technically, this requires some degree of regularity on the potential. But, is there a convincing ...
- 201
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Need help to fully understand SciPy's odeint's reported step sizes, eval times, # of funct calls & total proc. time (re. question in Astronomy SE)
A recent question in Astronomy SE Numerical Programming using odeint takes more than 17 minutes got me interested in looking closer at SciPy's odeint.
The problem is a modified orbital mechanical ...
- 1,016
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Velocity Verlet leading to faster simulation than Euler in an n-Body simulation?
I have all the constants set to the same values for each set of code, G, the timestep, the masses of the planets etc. But using Velocity Verlet doesn't work unless I lower the gravitational constant ...
- 95
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Why is velocity Verlet better than Verlet for gravity if it has a worse order of magnitude for the error term
Even though this method is more widely used than the simple Verlet method mentioned above, it unfortunately has an error term of O(Δt^2)
, which is two orders of magnitude worse. That said, if you ...
- 95
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0
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46
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Numerov Method with Time Varying Potential
Is it possible to use the Numerov method to solve the Time Dependent Schrodinger Equation ($\frac{i\partial\Psi(x, y, z, t)}{\partial t} = \nabla^2\Psi(x, y, z, t) + \Psi(x, y, z, t)V(x, y, z, t)$) ...
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Best approach to solve this system of equations?
I have the following 1D (in space, that is) system of equations I would like to solve:
\begin{equation}
\rho_{fs}\frac{\partial x_{fs}}{\partial t} = h_m\left(W_a - W_{fs}\right) - D_{eff}\left(\...
- 9
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100
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Stabilized Many Stage Runge-Kutta methods instead of Local/Multirate Time Stepping
Locally refined meshes are often inevitable for accurate, yet feasible computations.
In the context of time-dependent PDEs, however, this comes at the cost that (due to the CFL condition) reducing the ...
- 904
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248
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Passing additional arguments to `odeint` from `torchdiffeq` to solve an IVP
In Python I use the package torchdiffeq (as provided here) to solve initial value problems. Given an arbitrary function ...
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Using velocity verlet algorithm for nbody simulation results in planet leaving orbit
I'm new to computational physics and attempting an n-Body simulator in java but for some reason the planet in orbit leaves the orbit when it reaches the fastest accelaration aka the closest point to ...
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How does non-dimensionalization improve the behavior of ODE solvers?
I have a set of coupled ODEs that I'm solving numerically. The independent variable is time and runs from values of $10^{15}$ to $10^{17}$ in units of seconds. The state variables in their usual ...
- 193
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Solving the time dependent Schrödinger equation with leapfrog integration in 1D
To my frustration I am struggling to implement leapfrog integration for the time dependent Schrödinger equation.
To the best of my knowledge this was first explicitly done in "A fast explicit ...
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Time Integrators for Water Wave Simulation
I am interested in using a numerical wave tank (NWT) to study the performance of various water wave dampers using MATLAB. I am looking at nonlinear water waves. My current NWT (2d, periodic) is based ...
- 123
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100
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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 ...
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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
...
- 725
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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 ...
- 193
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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 ...
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Numerical evaluation of Duhamel's integration
I am trying to numerically evaluate the following Duhamel's integration:
$$
x = \frac{-1}{\omega_d} \int_0^t \ddot{x}_g (\tau) e^{-\zeta \omega_n(t - \tau)} \sin{\left( \omega_d (t - \tau) \right)} d\...
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Implicit integrator for ODE with quadratic right-hand side
I have an ODE for an unknown $x(t):[0,\infty)\to\mathbb R^n$ of the following form:
$$
x_i'(t)=a_i^\top x(t) + x(t)^\top Q_i x(t),
$$
for $i\in\{1,\ldots,n\}$. Here, the vectors $a_i\in\mathbb R^n$ ...
- 1,341
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votes
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answer
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Jos Stam's stable fluids — why is the timestep multiplied by the number of grid cells?
One more question about Jos Stam's GDC tutorial on stable fluids: in the advection step on page 8, the timestep for each dimension is implemented as dt * N, where <...
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239
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Write incompressible Navier Stokes as ODE in $(\mathbf{u},p)$
Consider the Navier stokes equation after the discretization with conforming finite elements with source term $f=0$. We have the algebraic structure of a saddle point problem:
$$M \dot{u} = f- Au -B^...
- 261
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votes
0
answers
91
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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^{-...
- 601
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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(...
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352
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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-...
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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 $a(\cdot)$ ...
- 1,341
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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
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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
$$\\\frac{\partial u}{\partial t} + C(x, t) \frac{\partial u}{\partial x} = 0 \\\\$$
$$ \\ C(x,t) = \...
- 1
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0
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Conservation of energy test for 2-body problem
I'm trying to implement a C++ code for the evaluation of the solution of an N-body system of ODE. I've started with a 2-body problem just to set the methods ...
- 31
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152
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Second Order Time Integration for Stiff Linear System Avoiding any Explicit Step
I have a linear system
$$
\dot x(t) = Ax(t), \quad x(0)=x_0 \tag{*}
$$
with $A$ being Hurwitz (i.e. the solutions may oscillate but will eventually tend to zero) but really stiff. $A$ might be large, ...
- 3,408
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Handling time derivative as source term in SDIRK methods
I am currently facing some challenge implementing a less traditional PDE which take a form similar to the Navier-Stokes equation, except that the continuity equation is modified such that:
$$\epsilon \...
- 1,147
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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) \...
- 152
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votes
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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 ...
- 101
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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)
$$
\Delta t \approx \frac{h}{N^2} \frac{C}{|u| + |c| + \frac
{N^2 \mu}{h}}
$$
($...
- 141
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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 ...
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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 &...
- 601
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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 ...
- 481
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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 ...
- 481
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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
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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 ...
3
votes
2
answers
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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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votes
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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 ...
- 143
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votes
1
answer
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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$$
$C$ is the heat capacity matrix. $K$ is the thermal conductivity matrix. $T$ is the temperature vector. $...
- 840
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136
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Lax-wendroff for stiff source terms
I am interested in problems of the form
$$
u_t = F(u) + S(u)
$$
where $F(u) = - div(f(u))$ and $S(u)$ is a stiff source term. I am looking for any existing works which develop Lax-Wendroff type ...
- 2,993
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vote
2
answers
66
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Numerical integral with symbolic integral in exponent
Many times in fourier approximation we come across integrals such as
$$\int_0^1 e^{-\gamma\int_0^xu_0(\eta)d\eta}dx$$ where $\gamma$ is a constant and the data for $u_0$ is provided as a discretely ...
- 149
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Time integration of wave equation
My question is: how come that certain formulations of the wave equation can be time integrated more efficiently then others?
Le me expand a bit on that. Consider the wave equation:
$$ \frac{d^2 p(t,...
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0
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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....
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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 ...
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