In this paper we study the convergence of the semi-implicit and the implicit Euler methods for the time integration of abstract, quasilinear hyperbolic evolution equations. The analytical framework considered here includes certain quasilinear Maxwell’s and wave equations as special cases. Our analysis shows that the Euler approximations are well-posed and convergent of order one. The techniques...

The system of isentropic Euler equations in the potential flow regime can be considered formally as a second order ordinary differential equation on the Wasserstein space of probability measures. This interpretation can be used to derive a variational time discretization. We prove that the approximate solutions generated by this discretization converge to a measure-valued solution of the isentr...

Abstract. Using the spectral dynamics, we study the critical threshold phenomena in the multidimensional restricted Euler (RE) equations. We identify sub-critical and sup-critical initial data for all space dimensions, which extends the previous result for the 3D and 4D restricted Euler equations. Our result suggests that: if the number of dimensions is odd, the finite time blowup is generic; i...

This is a brief overview of work done by Ian Anderson, Mark Fels, and myself on symmetry reduction of Lagrangians and Euler-Lagrange equations, a subject closely related to Palais’ Principle of Symmetric Criticality. After providing a little history, I describe necessary and sufficient conditions on a group action such that reduction of a group-invariant Lagrangian by the symmetry group yields ...

We describe a pseudo-spectral numerical method to solve the systems of one-dimensional evolution equations for free surface waves in a homogeneous layer of an ideal fluid. We use the method to solve a system of one-dimensional integro-differential equations, first proposed by Ovsjannikov and later derived by Dyachenko, Zakharov, and Kuznetsov, to simulate the exact evolution of nonlinear free s...

Given a fluid equation with reduced Lagrangian l which is a functional of velocity u and advected density D given in Eulerian coordinates, we give a general method for semidiscretising the equations to give a canonical Hamiltonian system; this system may then be integrated in time using a symplectic integrator. The method is Lagrangian, with the variables being a set of Lagrangian particle posi...

In this brief note we show that the author’s previous result in [1] on the nonexistence of self-similar singularities for the 3D incompressible Euler equations implies actually the nonexistence of ‘locally self-similar’ singular solution, which has been sought by many physicists. Nonexistence of locally self-similar solution We are concerned here on the following Euler equations for the homogen...

We study weak convergence of an Euler scheme for nonlinear stochastic delay differential equations (SDDEs) driven by multidimensional Brownian motion. The Euler scheme has weak order of convergence 1, as in the case of stochastic ordinary differential equations (SODEs) (i.e., without delay). The result holds for SDDEs with multiple finite fixed delays in the drift and diffusion terms. Although ...

We are concerned with the existence and stability of multidimensional transonic shocks for the Euler equations for steady potential compressible fluids. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for the velocity, can be written as the following second-order nonlinear equation of mixed elliptic-hyperbolic type for the velocity potential φ : Ω ⊂ R → R: ...

Explicit Runge–Kutta schemes with large stable step sizes are developed for integration of high order spectral difference spatial discretizations on quadrilateral grids. The new schemes permit an effective time step that is substantially larger than the maximum admissible time step of standard explicit Runge–Kutta schemes available in literature. Furthermore, they have a small principal error n...