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In this work, we adapt techniques of artificial and spectral viscosities [especially those presented in Klockner et al. (Math Model Nat Phenom 6(03):57–83, 2011) and Tadmor and Waagan (SIAM J Sci Comput 34(2):A993–A1009, 2012)] to develop adaptive modal filters for Legendre polynomial bases and demonstrate their application in a shock-capturing discontinuous Galerkin method. While traditional artificial viscosities can impose additional stability restrictions on the time step size, we show that our filtering methods can achieve a similar shock-capturing capability without additional restriction of the time step size. We further demonstrate that the accuracy of the artificial viscosity filter can be significantly improved upon with a hybrid spectral viscosity-artifical viscosity filter, which we also develop and demonstrate. We show that we can achieve computational efficiency with an analytical time integration technique similar to one presented in Moura et al. (Diffusion-based limiters for discontinuous galerkin methods-part I: one-dimensional equations, 2013). We consider several 1D numerical examples for the linear advection, scalar wave, Burgers’, and Euler equations to demonstrate the accuracy and uniform stability of our methods.

We propose a new particle-in-cell (PIC) method for the simulation of plasmas based on a recently developed, unconditionally stable solver for the wave equation. This method is not subject to a CFL restriction, limiting the ratio of the time step size to the spatial step size, typical of explicit methods, while maintaining computational cost and code complexity comparable to such explicit schemes. We describe the implementation in one and two dimensions for both electrostatic and electromagnetic cases, and present the results of several standard test problems, showing good agreement with theory with time step sizes much larger than allowed by typical CFL restrictions.

Building upon recent results obtained in Causley and Christlieb (SIAM J Numer Anal 52(1):220–235, 2014), Causley et al. (Math Comput 83(290):2763–2786, 2014, Method of lines transpose: high order L-stable O(N) schemes for parabolic equations using successive convolution, 2015), we describe an efficient second-order, unconditionally stable scheme for solving the wave equation, based on the method of lines transpose (MOL $$^T$$ ), and the resulting semi-discrete (i.e. continuous in space) boundary value problem. In Causley and Christlieb (SIAM J Numer Anal 52(1):220–235, 2014), unconditionally stable schemes of high order were derived, and in Causley et al. (Method of lines transpose: high order L-stable O(N) schemes for parabolic equations using successive convolution, 2015) a high order, fast $$\mathcal {O}(N)$$ spatial solver was derived, which is matrix-free and is based on dimensional-splitting. In this work, are interested in building a wave solver, and our main concern is the development of boundary conditions. We demonstrate all desired boundary conditions for a wave solver, including outflow boundary conditions, in 1D and 2D. The scheme works in a logically Cartesian fashion, and the boundary points are embedded into the regular mesh, without incurring stability restrictions, so that boundary conditions are imposed without any reduction in the order of accuracy. We demonstrate how the embedded boundary approach works in the cases of Dirichlet and Neumann boundary conditions. Further, we develop outflow and periodic boundary conditions for the MOL $$^T$$ formulation. Our solver is designed to couple with particle codes, and so special attention is also paid to the implementation of point sources, and soft sources which can be used to launch waves into waveguides.

When a periodic waveform with a discrete-harmonic spectrum is temporally windowed to make a signal, its spectrum becomes a continuous function of frequency. However, there are discrete-frequency representations for windowed signals such as the Fourier series representation of a periodically extended signal. This article introduces the concept of matching between the temporal window and the periodic waveform. Matching leads to a discrete-frequency representation in which the Fourier transform of the windowed signal preserves the amplitudes and phases of the waveform on the set of original waveform frequencies. Generating signals with matched window and waveform leads to important control of experiments.