Questions tagged [time-integration]

For questions about the particulars of solving differential equations with time as the independent variable.

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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 ...
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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 ...
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1 answer
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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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2 votes
1 answer
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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 $a_i\in\mathbb R^n$ ...
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3 votes
1 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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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^...
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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^{-...
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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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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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1 answer
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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 $a(\cdot)$ ...
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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 ...
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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) = \...
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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 ...
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1 answer
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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, ...
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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 \...
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1 answer
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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) \...
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2 votes
1 answer
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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 ...
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3 votes
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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) $$ \Delta t \approx \frac{h}{N^2} \frac{C}{|u| + |c| + \frac {N^2 \mu}{h}} $$ ($...
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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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2 answers
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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 &...
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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 ...
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  • 471
3 votes
1 answer
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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 ...
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  • 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 ...
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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 ...
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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} ...
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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 ...
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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. $...
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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 ...
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2 answers
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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 ...
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  • 149
3 votes
1 answer
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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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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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  • 216
1 vote
1 answer
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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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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{\...
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1 answer
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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 ...
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  • 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 ...
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  • 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 ...
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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 ...
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  • 379
1 vote
1 answer
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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 ...
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  • 321
1 vote
1 answer
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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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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 ...
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1 answer
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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 ...
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  • 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 ...
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3 votes
2 answers
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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, ...
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  • 163
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2 answers
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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 ...
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  • 141
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0 answers
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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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  • 83
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0 answers
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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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  • 533
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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 ...
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  • 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) =...
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