Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014

We consider the C0 interior penalty Galerkin method for biharmonic eigenvalue problems with the boundary conditions of the clamped plate, the simply supported plate and the Cahn-Hilliard type. We establish the convergence of the method and present numerical results to illustrate its performance. We also compare it with the Argyris C1 finite element method, the Ciarlet-Raviart mixed finite element method, and the Morley nonconforming finite element method.

Strong stability preserving (SSP) high order time discretizations were developed to address the need for nonlinear stability properties in the numerical solution of hyperbolic partial differential equations with discontinuous solutions. These methods preserve the monotonicity properties (in any norm, seminorm or convex functional) of the spatial discretization coupled with first order Euler time stepping. This review paper describes the state of the art in SSP methods.

We examine the use of Hermite interpolation, that is interpolation using derivative data, in place of Lagrange interpolation to develop high-order PDE solvers. The fundamental properties of Hermite interpolation are recalled, with an emphasis on their smoothing effect and robust performance for nonsmooth functions. Examples from the CHIDES library are presented to illustrate the construction and performance of Hermite methods for basic wave propagation problems.

We design adaptive high-order Galerkin methods for the solution of linear elliptic problems and study their performance. We first consider adaptive Fourier-Galerkin methods and Legendre-Galerkin methods, which offer unlimited approximation power only restricted by solution and data regularity. Their analysis of convergence and optimality properties reveals a sparsity degradation for Gevrey classes. We next turn our attention to the h p-version of the finite element method, design an adaptive scheme which hinges on a recent algorithm by P. Binev for adaptive h p-approximation, and discuss its optimality properties.

We present a numerical framework for simulation of the compressible Navier-Stokes equations on problems with deforming domains where the boundary motion is prescribed by moving meshes. Our goal is a high-order accurate, efficient, robust, and general purpose simulation tool. To obtain this, we use a discontinuous Galerkin space discretization, diagonally implicit Runge-Kutta time integrators, and fully unstructured meshes of triangles and tetrahedra. To handle the moving boundaries, a mapping function is produced by first deforming the mesh using a neo-Hookean elasticity model and a high-order continuous Galerkin FEM method. The resulting nonlinear equations are solved using Newton’s method and a robust homotopy approach. From the deformed mesh, we compute grid velocities and deformations that are consistent with the time integration scheme. These are used in a mapping-based arbitrary Lagrangian-Eulerian formulation, with numerically computed mapping Jacobians which satisfy the geometric conservation law. We demonstrate our methods on a number of problems, ranging from model problems that confirm the high-order accuracy to the flow in domains with complex deformations.

There has been much work in the area of superconvergent error analysis for finite element and discontinuous Galerkin (DG) methods. The property of superconvergence leads to the question of how to exploit this information in a useful manner, mainly through superconvergence extraction. There are many methods used for superconvergence extraction such as projection, interpolation, patch recovery and B-spline convolution filters. This last method falls under the class of Smoothness-Increasing Accuracy-Conserving (SIAC) filters. It has the advantage of improving both smoothness and accuracy of the approximation. Specifically, for linear hyperbolic equations it can improve the order of accuracy of a DG approximation from k + 1 to 2k + 1, where k is the highest degree polynomial used in the approximation, and can increase the smoothness to k − 1. In this article, we discuss the importance of overcoming the mathematical barriers in making superconvergence extraction techniques useful for applications, specifically focusing on SIAC filtering.

This article centers on the computational performance of the continuous and discontinuous Galerkin time stepping schemes for general first-order initial value problems in $$\mathbb{R}^{n}$$ , with continuous nonlinearities. We briefly review a recent existence result for discrete solutions from Janssen and Wihler (Existence results for the continuous and discontinuous Galerkin time stepping methods for nonlinear initial value problems, 2014, Submitted), and provide a numerical comparison of the two time discretization methods.

In this paper, we consider the problem of reconstructing piecewise smooth functions to high accuracy from nonuniform samples of their Fourier transform. We use the framework of nonuniform generalized sampling (NUGS) to do this, and to ensure high accuracy we employ reconstruction spaces consisting of splines or (piecewise) polynomials. We analyze the relation between the dimension of the reconstruction space and the bandwidth of the nonuniform samples, and show that it is linear for splines and piecewise polynomials of fixed degree, and quadratic for piecewise polynomials of varying degree.

We develop a high performance computing (HPC) framework for efficient simulations of a class of fractional-order partial differential equations (FPDE), using high-order in time and space parallel algorithms. HPC systems provide a large number of processing cores with limitations on the amount of memory available per core. Such limitations impose severe constraints for resolving fine spatial structures that require large degrees of freedom (DoF). In this article, using several message passing interface (MPI) communicators, we develop and demonstrate an efficient hybrid framework that combines parallel in time and space tasks that facilitate careful balance between parallel performance within the memory constraint to simulate the FPDE model. We demonstrate the approach for a 3D fractional PDE using several million spatial DoF.

In this work we discuss a newly developed class of robust and high-order accurate upwind schemes for wave equations in second-order form on curvilinear and overlapping grids. The schemes are based on embedding d’Alembert’s exact solution for a local Riemann-type problem directly into the discretization (Banks and Henshaw, J Comput Phys 231(17):5854–5889, 2012). High-order accuracy is obtained using a single-step space-time scheme. Overlapping grids are used to represent geometric complexity. The method of manufactured solutions is used to demonstrate that the dissipation introduced through upwinding is sufficient to stabilize the wave equation in the presence of overlapping grid interpolation.

A robust interface treatment for the discontinuous coefficient advection equation satisfying time-independent jump conditions is presented. The aim of the investigation is to show how the different concepts like well-posedness, conservation and stability are related. The equations are discretized using high order finite difference methods on Summation By Parts (SBP) form. The interface conditions are weakly imposed using the Simultaneous Approximation Term (SAT) procedure. Spectral analysis and numerical simulations corroborate the theoretical findings.

Filtering is necessary to stabilize piecewise smooth solutions. The resulting diffusion stabilizes the method, but may fail to resolve the solution near discontinuities. Moreover, high order filtering still requires cost prohibitive time stepping. This paper introduces an adaptive filter that controls spurious modes of the solution, but is not unnecessarily diffusive. Consequently we are able to stabilize the solution with larger time steps, but also take advantage of the accuracy of a high order filter.

Typically when a semi-discrete approximation to a partial differential equation (PDE) is constructed a discretization of the spatial operator with a truncation error τ is derived. This discrete operator should be semi-bounded for the scheme to be stable. Under these conditions the Lax–Richtmyer equivalence theorem assures that the scheme converges and that the error will be, at most, of the order of $$\|\tau \|$$ . In most cases the error is in indeed of the order of $$\|\tau \|$$ . We demonstrate that for the Heat equation stable schemes can be constructed, whose truncation errors are τ, however, the actual errors are much smaller. This gives more degrees of freedom in the design of schemes which can make them more efficient (more accurate or compact) than standard schemes. In some cases the accuracy of the schemes can be further enhanced using post-processing procedures.

The performance of a hybrid compact (Compact) finite difference scheme and characteristic-wise weighted essentially non-oscillatory (WENO) finite difference scheme (Hybrid) for the detonation waves simulations is investigated. The Hybrid scheme employs the nonlinear 5th-order WENO-Z scheme to capture high gradients and discontinuities in an essentially non-oscillatory manner and the linear 6th-order Compact scheme to resolve the fine scale structures in the smooth regions of the solution in an efficient and accurate manner. Numerical oscillations generated by the Compact scheme is mitigated by the high order filtering. The high order multi-resolution algorithm is employed to detect the smoothness of the solution. The Hybrid scheme allows a potential speedup up to a factor of three or more for certain classes of shocked problems. The simulations of one-dimensional shock-entropy wave interaction and classical stable detonation waves, and the two-dimensional detonation diffraction problem around a 90∘ corner show that the Hybrid scheme is more efficient, less dispersive and less dissipative than the WENO-Z scheme.

Experiences from using a higher order accurate finite difference multiblock solver to compute the time dependent flow over a cavity is summarized. The work has been carried out as part of a work in a European project called IDIHOM in a collaboration between the Swedish Defense Research Agency (FOI) and University of Linköping (LiU). The higher order code is based on Summation By Parts operators combined with the Simultaneous Approximation Term approach for boundary and interface conditions. The spatial accuracy of the code is verified by calculations over a cyclinder by monitoring the decay of the errors of known wall quantities as the grid is refined. The focus is on the validation for a test case of transonic flow over a rectangular cavity with hybrid RANS/LES calculations. The results are compared to reference numerical results from a second order finite volume code as well as with experimental results with a good overall agreement between the results.

Designing numerical methods with high-order accuracy for problems in irregular domains and/or with interfaces is crucial for the accurate solution of many problems with physical and biological applications. The major challenge here is to design an efficient and accurate numerical method that can capture certain properties of analytical solutions in different domains/subdomains while handling arbitrary geometries and complex structures of the domains. Moreover, in general, any standard method (finite-difference, finite-element, etc.) will fail to produce accurate solutions to interface problems due to discontinuities in the model’s parameters/solutions. In this work, we consider Difference Potentials Method (DPM) as an efficient and accurate solver for the variable coefficient elliptic interface problems.

This paper describes extensions of the generalized summation-by-parts (GSBP) framework to the approximation of the second derivative with a variable coefficient and to time integration. GSBP operators for the second derivative lead to more efficient discretizations, relative to the classical finite-difference SBP approach, as they can require fewer nodes for a given order of accuracy. Similarly, for time integration, time-marching methods based on GSBP operators can be more efficient than those based on classical SBP operators, as they minimize the number of solution points which must be solved simultaneously. Furthermore, we demonstrate the link between GSBP operators and Runge-Kutta time-marching methods.

We present a scheme to solve three-dimensional viscoelastic anisotropic wave propagation on structured staggered grids. The scheme uses a fully-staggered grid (FSG) or Lebedev grid (Lebedev, J Sov Comput Math Math Phys 4:449–465, 1964; Rubio et al. Comput Geosci 70:181–189, 2014), which allows for arbitrary anisotropy as well as grid deformation. This is useful when attempting to incorporate a bathymetry or topography in the model. The correct representation of surface waves is achieved by means of using high-order mimetic operators (Castillo and Grone, SIAM J Matrix Anal Appl 25:128–142, 2003; Castillo and Miranda, Mimetic discretization methods. CRC Press, Boca Raton, 2013), which allow for an accurate, compact and spatially high-order solution at the physical boundary condition. Furthermore, viscoelastic attenuation is represented with a generalized Maxwell body approximation, which requires of auxiliary variables to model the convolutional behavior of the stresses in lossy media. We present the scheme’s accuracy with a series of tests against analytical and numerical solutions. Similarly we show the scheme’s performance in high-performance computing platforms. Due to its accuracy and simple pre- and post-processing, the scheme is attractive for carrying out thousands of simulations in quick succession, as is necessary in many geophysical forward and inverse problems both for the industry and academia.

We present a locally conservative spectral least-squares formulation for the scalar diffusion-reaction equation in curvilinear coordinates. Careful selection of a least squares functional and compatible finite dimensional subspaces for the solution space yields the conservation properties. Numerical examples confirm the theoretical properties of the method.

Atmospheric flows are characterized by a large range of length scales as well as strong gradients. The accurate simulation of such flows requires numerical algorithms with high spectral resolution, as well as the ability to provide nonoscillatory solutions across regions of high gradients. These flows exhibit a large range of time scales as well—the slowest waves propagate at the flow velocity and the fastest waves propagate at the speed of sound. Time integration with explicit methods are thus inefficient, although algorithms with semi-implicit time integration have been used successfully in past studies. We propose a finite-difference method for atmospheric flows that uses a weighted compact scheme for spatial discretization and implicit-explicit additive Runge-Kutta methods for time integration. We present results for a benchmark atmospheric flow problem and compare our results with existing ones in the literature.

Recent developments in the SBP-SAT method have made available high-order interpolation operators (Mattsson and Carpenter, SIAM J Sci Comput 32(4):2298–2320, 2010). Such operators allow the coupling of different SBP methods across nonconforming interfaces of multiblock grids while retaining the three fundamental properties of the SBP-SAT method: strict stability, accuracy, and conservation. As these interpolation operators allow a more flexible computational mesh, they are appealing for complex geometries. Moreover, they are well suited for problems involving sliding meshes, like rotor/stator interactions, wind turbines, helicopters, and turbomachinery simulations in general, since sliding interfaces are (almost) always nonconforming. With such applications in mind, this paper presents an accuracy analysis of these interpolation operators when applied to fluid dynamics problems on moving grids. The classical problem of an inviscid vortex transported by a uniform flow is analyzed: the flow is governed by the unsteady Euler equations and the vortex crosses a sliding interface. Furthermore, preliminary studies on a rotor/stator interaction are also presented.

This note describes an implementation of a discontinuous Petrov Galerkin (DPG) method for acoustic waves within the framework of high order finite elements provided by the software package NGSolve. A technique to impose the impedance boundary condition weakly is indicated. Numerical results from this implementation show that a multiplicative Schwarz algorithm, with no coarse solve, provides a p-preconditioner for solving the DPG system. The numerical observations suggest that the condition number of the preconditioned system is independent of the frequency k and the polynomial degree p.

Recent gain of interest in discontinuous Galerkin (DG) methods shows their success in computational fluid dynamics. One potential drawback is the high number of globally coupled unknowns. By means of hybridization, this number can be significantly reduced. The hybridized DG (HDG) method has proven to be beneficial especially for steady flows. In this work we apply it to a time-dependent flow problem with shocks. Due to its inherently implicit structure, time integration methods such as diagonally implicit Runge-Kutta (DIRK) methods present themselves as natural candidates. Furthermore, as the application of flux limiting to HDG is not straightforward, an artificial viscosity model is applied to stabilize the method.

The purpose of this paper is to extend a novel numerical methodology, combining thermal immersed boundary and Fourier pseudospectral methods called IMERSPEC. This methodology has been developed for incompressible fluid flow problems modeled using Navier-Stokes, mass and energy equations. The numerical algorithm consists of Fourier pseudospectral method (FPSM), where Dirichlet boundary condition is modeled using an immersed boundary method (multi-direct forcing method). The new method combines the advantages of high accuracy and low computational cost provided by FPSM to the possibility of managing complex and non periodical geometries given by immersed boundary method. In the present work this new methodology is applied to the problem of heat transfer for natural convection in the annulus between horizontal concentric cylinders and conducted to validate the capability and efficiency of present method. Results for this application are presented and good agreement with available data in the literature have been achieved.

In Kotov et al. (Proceedings of ICCFD8, 2014) the LES of a turbulent flow with a strong shock by Yee and Sjögreen (Proceedings of ICOSAHOM 09, Trondheim, Norway, 2013) scheme indicated a good agreement with the filtered DNS data. There are vastly different requirements in the minimization of numerical dissipation for accurate turbulence simulations of different compressible flow types and flow speeds. The present study examines the versatility of the Yee and Sjögreen scheme for LES of low speed flows. Special attention is focused on the accuracy performance of this scheme using the Smagorinsky and the Germano-Lilly SGS models.

In this contribution we present a sub-cell discretization method for the computation of the interface velocities involved in the convective terms of the incompressible Navier-Stokes equations. We compute an interface velocity by solving a local two-point boundary value problem (BVP) iteratively. To account for the two-dimensionality of the interface velocity we introduce a constant cross-flux term in our computation. The discretization scheme is used to simulate the flow in a lid-driven cavity.

In the description of water waves, dispersion is one of the most important physical properties; it specifies the propagation speed as function of the wavelength. Accurate modelling of dispersion is essential to obtain high-quality wave propagation results. The relation between speed and wavelength is given by a non-algebraic relation; for finite element/difference methods this relation has to be approximated and leads to restrictions for waves that are propagated correctly. By using a spectral implementation dispersion can be dealt with exactly above flat bottom using a pseudo-differential operator so that all wavelengths can be propagated correctly. However, spectral methods are most commonly applied for problems in simple domains, while most water wave applications need complex geometries such as (harbour) walls, varying bathymetry, etc.; also breaking of waves requires a local procedure at the unknown position of breaking. This paper deals with such inhomogeneities in space; the models are formulated using Fourier integral operators and include non-trivial localization methods. The efficiency and accuracy of a so-called spatial-spectral implementation is illustrated here for a few test cases: wave run-up on a coast, wave reflection at a wall and the breaking of a focussing wave. These methods are included in HAWASSI software (Hamiltonian Wave-Ship-Structure Interaction) that has been developed over the past years.

We sketch a modal tau approach for treating binary neutron stars, in particular a low-rank technique for dealing with the changing surface of a tidally distorted star.

A characterisation theorem for best uniform wavenumber approximations by central difference schemes is presented. A central difference stencil is derived based on the theorem and is compared with dispersion relation preserving schemes and with classical central differences for a relevant test problem.

This paper presents a curved meshing technique for unstructured tetrahedral meshes where G1 surface continuity is maintained for the triangular element faces representing the curved domain surfaces. A bottom-up curving approach is used to support geometric models with multiple surface patches where either C0 or G1 geometry continuity between patches is desired. Specific parametrization approaches based on Bézier forms and blending functions are used to define the mapping for curved element faces and volumes between parametric and physical coordinate systems. A preliminary result demonstrates that using G1-continuity meshes can improve the solution results obtained.

We make an initial investigation into the temporal efficiency of a fully discrete summation-by-parts approach for stiff unsteady flows with boundary layers. As a model problem for the Navier–Stokes equations we consider a two-dimensional advection-diffusion problem with a boundary layer. The problem is discretized in space using finite difference approximations on summation-by-parts form together with weak boundary conditions, leading to optimal stability estimates. For the time integration part we consider various forms of high order summation-by-parts operators, and compare the results to an existing popular fourth order diagonally implicit Runge-Kutta method. To solve the resulting fully discrete equation system, we employ a multi-grid scheme with dual time stepping.

In the context of stabilization of high order spectral elements, we introduce a dissipative scheme based on the solution of the compressible Euler equations that are regularized through the addition of a residual-based stress tensor. Because this stress tensor is proportional to the residual of the unperturbed equations, its effect is close to none where the solution is sufficiently smooth, whereas it increases elsewhere. This paper represents a first extension of the work by Nazarov and Hoffman (Int J Numer Methods Fluids 71:339–357, 2013) to high-order spectral elements in the context of low Mach number atmospheric dynamics. The simulations show that the method is reliable and robust for problems with important stratification and thermal processes such as the case of moist convection. The results are partially compared against a Smagorinsky solution. With this work we mean to make a step forward in the implementation of a stabilized, high order, spectral element large eddy simulation (LES) model within the Nonhydrostatic Unified Model of the Atmosphere, NUMA.

High-order finite-difference methods are commonly used in wave propagator for industrial subsurface imaging algorithms. Computational aspects of the reduced linear elastic vertical transversely isotropic propagator are considered. Thread parallel algorithms suitable for implementing this propagator on multi-core and many-core processing devices are introduced. Portability is addressed through the use of the OCCA runtime programming interface. Finally, performance results are shown for various architectures on a representative synthetic test case.

In this paper we present an assessment of the discontinuous Galerkin (DG) formulation through modified equation analysis (MEA). When applied to linear advection, MEA can help to clarify wave-propagation properties previously observed in DG. In particular, a connection between MEA and dispersion-diffusion (eigensolution) analysis is highlighted. To the authors’ knowledge this is the first application of MEA to DG schemes, and as such this study focuses only on element-wise constant and linear discretizations in one dimension. For the linear discretization, we found that the physical mode’s accuracy can be increased via upwinding. MEA’s application to higher order solutions and non-linear problems is also briefly discussed. In special, we point out that MEA’s applicability in the analysis of DG-based implicit large eddy simulations seems infeasible due to convergence issues.

A time-dependent coordinate transformation of a constant coefficient hyperbolic system of equations is considered. We use the energy method to derive well-posed boundary conditions for the continuous problem. Summation-by-Parts (SBP) operators together with a weak imposition of the boundary and initial conditions using Simultaneously Approximation Terms (SATs) guarantee energy-stability of the fully discrete scheme. We construct a time-dependent SAT formulation that automatically imposes the boundary conditions, and show that the numerical Geometric Conservation Law (GCL) holds. Numerical calculations corroborate the stability and accuracy of the approximations. As an application we study the sound propagation in a deforming domain using the linearized Euler equations.

We consider the Saint Venant system (shallow water equations), i.e. an approximation of the incompressible Euler equations widely used to describe river flows, flooding phenomena or erosion problems. We focus on problems involving dry-wet transitions and propose a solution technique using the Spectral Element Method (SEM) stabilized with a variant of the Entropy Viscosity Method (EVM) that is adapted to treat dry zones.

A windowed Fourier method is proposed for approximation of non-periodic functions on equispaced nodes. Spectral convergence is obtained in most of the domain, except near the boundaries, where polynomial least-squares is used to correct the approximation. Because the method can be implemented using partition of unit and domain decomposition, it is suitable for adaptive and parallel implementations and large scale computations. Computations can be carried out using fast Fourier transforms. Comparisons with Fourier extension, rational interpolation and least-squares methods are presented.

It has been noted in the past that discontinuous Galerkin methods can be viewed as a low order multi-domain Spectral method with penalty term (Hesthaven et al., Spectral methods for time-dependent problems, Cambridge University Press, Cambridge, 2007). It is then logical to first ask how to relate filters in Spectral Methods to Smoothness-Increasing Accuracy-Conservin (SIAC) filters, which are typically applied to approximations obtained via the discontinuous Galerkin methods. In this article we make a first effort to relate Smoothness-Increasing Accuracy-Conserving filtering to filtering for Spectral Methods. We frame this discussion in the context of Vandeven (J Sci Comput 6:159–192, 1991).

We present an algorithm that reformulates existing methods to construct higher-order mimetic differential operators. Constrained linear optimization is the key idea of this resulting algorithm. The authors exemplified this algorithm by constructing an eight-order-accurate one-dimensional mimetic divergence operator. The algorithm computes the weights that impose the mimetic condition on the constructed operator. However, for higher orders, the computation of valid weights can only be achieved through this new algorithm. Specifically, we provide insights on the computational implementation of the proposed algorithm, and some results of its application in different test cases. Results show that for all of the proposed test cases, the proposed algorithm effectively solves the problem of computing valid weights, thus constructing higher-order mimetic operators.

For functions u ∈ H1(Ω) in an open, bounded polyhedron $$\varOmega \subset \mathbb{R}^{d}$$ of dimension d = 1, 2, 3, which are analytic in $$\overline{\varOmega }\setminus \mathcal{S}$$ with point singularities concentrated at the set $$\mathcal{S}\subset \overline{\varOmega }$$ consisting of a finite number of points in $$\overline{\varOmega }$$ , the exponential rate $$\exp (-b\root{d + 1}\of{N})$$ of convergence of h p-version continuous Galerkin finite element methods on families of regular, simplicial meshes in Ω can be achieved. The simplicial meshes are assumed to be geometrically refined towards $$\mathcal{S}$$ and to be shape regular, but are otherwise unstructured.

We review recent results on dimension-robust higher order convergence rates of Quasi-Monte Carlo Petrov-Galerkin approximations for response functionals of infinite-dimensional, parametric operator equations which arise in computational uncertainty quantification.

We develop stable finite difference approximations for a multi-physics problem that couples elastic wave propagation in one domain to acoustic wave propagation in another domain. The approximation consists of one finite difference scheme in each domain together with discrete interface conditions that couple the two schemes. The finite difference approximations use summation-by-parts (SBP) operators, which lead to stability of the coupled problem. Furthermore, we develop a new way to enforce boundary conditions for SBP discretizations of first order problems. The new method, which uses ghost points to enforce the boundary conditions, is a flexible alternative to the more established projection and SAT methods.

We propose and numerically investigate two approaches for extending the application area of transparent boundary conditions (TBCs) for the wave equation: a method for generating finite-difference approximations of TBCs with the fourth and sixth order in space, and a coupling procedure of TBCs on the top boundary of a cubical computational domain with characteristic BCs at the neighbor side boundaries.

In this work we compare different families of nested quadrature points, i.e. the classic Clenshaw–Curtis and various kinds of Leja points, in the context of the quasi-optimal sparse grid approximation of random elliptic PDEs. Numerical evidence suggests that both families perform comparably within such framework.

This chapter presents a constructive derivation of HDG methods for convection-diffusion-reaction equation using the Rankine-Hugoniot condition. This is possible due to the fact that, in the first order form, convection-diffusion-reaction equation is a hyperbolic system. As such it can be discretized using the standard upwind DG method. The key is to realize that the Rankine-Hugoniot condition naturally provides an upwind HDG framework. The chief idea is to first break the uniqueness of the upwind flux across element boundaries by introducing single-valued new trace unknowns on the mesh skeleton, and then re-enforce the uniqueness via algebraic conservation constraints. Essentially, the HDG framework is a redesign of the standard DG approach to reduce the number of coupled unknowns. In this work, an upwind HDG method with one trace unknown is systematically constructed, and then extended to a family of penalty HDG schemes. Various existing HDG methods are rediscovered using the proposed framework.

The present work proposes the extension of the IMERSPEC methodology for numerical simulations of two-phase flows. This methodology consists of the fusion between the Fourier pseudospectral method and the immersed boundary method for non-periodical problems. This method was originally developed for single-phase and incompressible flows (Mariano et al., Comput Model Eng Sci 59:181–216, 2010). In the present paper, we extend this methodology for two-phase flows using the front-tracking method to model the fluid-fluid interface. The results involving the spurious currents, mass conservation and analysis through numerical experimental bubbles rise, show that the proposed method can be considered validated and promising to computational fluid dynamics (CFD).

In general, solutions of nonlinear hyperbolic PDEs contain shocks or develop discontinuities. One option for improving the numerical treatment of the spurious oscillations that occur near these artifacts is through the application of a limiter. The cells where such treatment is necessary are referred to as troubled cells. In this article, we discuss the multiwavelet troubled-cell indicator that was introduced by Vuik and Ryan (J Comput Phys 270:138–160, 2014). We focus on the relation between the highest-level multiwavelet coefficients and jumps in (derivatives of) the DG approximation. Based on this information, we slightly modify the original multiwavelet troubled-cell indicator. Furthermore, we show one-dimensional test cases using the modified multiwavelet troubled-cell indicator.

We outline and extend results for an explicit local time stepping (LTS) strategy designed to operate with the discontinuous Galerkin spectral element method (DGSEM). The LTS procedure is derived from Adams-Bashforth multirate time integration methods. The new results of the LTS method focus on parallelization and reformulation of the LTS integrator to maintain conservation. Discussion is focused on a moving mesh implementation, but the procedures remain applicable to static meshes. In numerical tests, we demonstrate the strong scaling of a parallel, LTS implementation and compare the scaling properties to a parallel, global time stepping (GTS) Runge-Kutta implementation. We also present time-step refinement studies to show that the redesigned, conservative LTS approximations are spectrally accurate in space and have design temporal accuracy.