We present a space-time Trefftz discontinuous Galerkin method for approximating the acoustic wave equation semi-explicitly on tent pitched meshes. DG Trefftz methods use discontinuous test and trial functions, which solve the wave equation locally. Tent pitched meshes allow to solve the equation elementwise, allowing locally optimal advances in time. The method is implemented in NGSolve, solving the space-time elements in parallel, whenever possible. Insights into the implementational details are given, including the case of propagation in heterogeneous media.Mathematics Subject Classification: 65M60, 41A10, 35L05

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In a previous paper (Woods 1962) a theory is presented which leads to a general dispersion relation for hydromagnetic waves in a dissipative plasma contained in a cylindrical tube. In the present paper experimental observations are compared with the predictions of this theory.An experiment is described in which torsional hydromagnetic waves are excited in a gas discharge. Measurements of the wave velocity and damping are compared with solutions of the dispersion equation which are computed from measured values of the plasma parameters. The results are consistent with the theory, and good numerical agreement is obtained by assuming a loss of particles to the tube walls. There is evidence of a cut-off in wave propagation in the region of ion cyclotron resonance.

Trefftz methods are high-order Galerkin schemes in which all discrete functions are elementwise solution of the PDE to be approximated. They are viable only when the PDE is linear and its coefficients are piecewise-constant. We introduce a “quasi-Trefftz” discontinuous Galerkin (DG) method for the discretisation of the acoustic wave equation with piecewise-smooth material parameters: the discrete functions are elementwise approximate PDE solutions. We show that the new discretisation enjoys the same excellent approximation properties as the classical Trefftz one, and prove stability and high-order convergence of the DG scheme. We introduce polynomial basis functions for the new discrete spaces and describe a simple algorithm to compute them. The technique we propose is inspired by the generalised plane waves previously developed for time-harmonic problems with variable coefficients; it turns out that in the case of the time-domain wave equation under consideration the quasi-Trefftz approach allows for polynomial basis functions.

The head-on interaction, in the one-dimensional, unsteady isentropic flow of a perfect gas, of a simple compression wave and a simple expansion wave is studied by considering typical examples. The physical aspect of the problem is discussed in (1); in this note the possibility of shock formation is ignored, and the correspondence defined by the complete mathematical solution of the equations of isentropic flow between the x, t-plane and the plane of the characteristic variables is elucidated.The solution is distinguished by the appearance of two limit lines and a second-order limit point where they meet. It is found that the image of the characteristic plane in the x, t-plane is four-sheeted; all sheets overlap each other, but each covers only part of the plane, and the only point common to all sheets is the second-order limit point, where both limit lines are cusped (§ 3·1).The solution also contains an edge of regression, and a discussion of the properties of this type of singularity will be found in §§ 2 and 2·1.