149 research outputs found
Convergence analysis of some tent-based schemes for linear hyperbolic systems
Finite element methods for symmetric linear hyperbolic systems using
unstructured advancing fronts (satisfying a causality condition) are considered
in this work. Convergence results and error bounds are obtained for mapped tent
pitching schemes made with standard discontinuous Galerkin discretizations for
spatial approximation on mapped tents. Techniques to study semidiscretization
on mapped tents, design fully discrete schemes, prove local error bounds, prove
stability on spacetime fronts, and bound error propagated through unstructured
layers are developed
Structure Aware RungeâKutta Time Stepping for Spacetime Tents
We introduce a new class of RungeâKutta type methods suitable for time stepping to propagate hyperbolic solutions within tent-shaped spacetime regions. Unlike standard RungeâKutta methods, the new methods yield expected convergence properties when standard high order spatial (discontinuous Galerkin) discretizations are used. After presenting a derivation of nonstandard order conditions for these methods, we show numerical examples of nonlinear hyperbolic systems to demonstrate the optimal convergence rates. We also report on the discrete stability properties of these methods applied to linear hyperbolic equations
Stability of Structure-Aware Taylor Methods for Tents
Structure-aware Taylor (SAT) methods are a class of timestepping schemes designed for propagating linear hyperbolic solutions within a tent-shaped spacetime region. Tents are useful to design explicit time marching schemes on unstructured advancing fronts with built-in locally variable timestepping for arbitrary spatial and temporal discretization orders. The main result of this paper is that an s-stage SAT timestepping within a tent is weakly stable under the time step constraint
ât †Ch1+1/s , where ât is the time step size and h is the spatial mesh size. Improved stability properties are also presented for high-order SAT time discretizations coupled with low-order spatial polynomials. A numerical verification of the sharpness of proven estimates is also included
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Computational Engineering
The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods
A cVEM-DG space-time method for the dissipative wave equation
A novel space-time discretization for the (linear) scalar-valued dissipative wave equation is presented. It is a structured approach, namely, the discretization space is obtained tensorizing the Virtual Element (VE) discretization in space with the Discontinuous Galerkin (DG) method in time. As such, it combines the advantages of both the VE and the DG methods. The proposed scheme is implicit and it is proved to be unconditionally stable and accurate in space and time
A space-time quasi-Trefftz DG method for the wave equation with piecewise-smooth coefficients
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 method for the
discretisation of the acoustic wave equation with piecewise-smooth wavespeed:
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.Comment: 25 pages, 9 figure
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Computational Engineering
This Workshop treated a variety of finite element methods and applications in computational engineering and expanded their mathematical foundation in engineering analysis. Among the 53 participants were mathematicians and engineers with focus on mixed and nonstandard finite element schemes and their applications
Asynchronous parallel solver for hyperbolic problems via the Spacetime Discontinuous Galerkin method
This thesis presents a parallel Space Time Discontinuous Galerkin (SDG) finite element method which makes use of the method's unstructured mesh generation and localized solution technique to achieve a high level of parallel scalability. Our SDG method is different from most traditional adaptive finite element methods in that the solution process generates fully unstructured spacetime grids that satisfy a special causality constraint ensuring that computations can occur locally on small cluster of spacetime elements. The resulting asynchronous solution scheme offers several desirable features: element-wise conservation of solution quantities, strong stability properties without the need for explicit stabilization, local mesh adaptivity operations and linear complexity in the number of spacetime elements.
In this thesis we propose an algorithm that effectively parallelizes the Tent Pitcher algorithm developed by [1] using the POSIX Thread (or Pthread) parallel execution model. Multiple software threads can simultaneously and asynchronously perform patch computations by advancing vertices in time. By enforcing the causality constraint on the time step, we can guarantee that each thread only performs calculations using data computed previously. Additionally, improvements to the adaptivity scheme allow for local mesh refinement and coarsening while maintaining globally conforming triangulation. Numerical tests show that our algorithm achieves high parallel scalability using shared-memory parallelization
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