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# Questions tagged [time-integration]

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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 ...
32 views

### Are 2nd spatial derivatives useful for integrating ODEs?

In discussion of adaptive integrators for ODEs, I see a lot of discussion of how second derivatives in time can be approximated using finite differences, i.e., take several steps, and use numerical ...
47 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 ...
438 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 ...
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### 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 ...
228 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 ...
49 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: ...
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. ...
153 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....
54 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 ...
14k views

### How to formulate lumped mass matrix in FEM

When solving time dependent PDE's using the finite element method, for example say the heat equation, if we use explicit time stepping then we have to solve a linear system because of the mass matrix. ...
252 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: ...
235 views

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### 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 ...
117 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 ...
858 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 ...
221 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 ...
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### 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 ...
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### Computational time not proportional to integration interval in ODE-solver?

I am running octave and i have been trying ode45, ode54, ode23 etc to integrate the equation ` $$Q''(t) = B\cos(Q)\sin(\omega t)$$ $$Q(t=0)=0.$$ When the time interval to be integrated increases, the ...