Local Space-Time Adaptive Discontinuous Galerkin Finite Element Methods for Time-Dependent Waves

2003 ◽  
Author(s):  
Lonny L. Thompson ◽  
Dantong He
Keyword(s):  
Discontinuous Galerkin ◽  
Acoustic Radiation ◽  
Adaptive Strategy ◽  
Adaptive Methods ◽  
Space Time ◽  
Time Dependent ◽  
Time Interval ◽  
Time Step ◽  
Continuous Space ◽  
Local Space

Local space-time adaptive methods are developed including high-order accurate nonreflecting boundary conditions (NRBC) for time-dependent waves. The time-discontinuous Galerkin (TDG) variational method is used to divide the time-interval into space-time slabs, the solution advanced from one slab to the next. Within each slab, a continuous space-time mesh is used which enables local sub-time steps. By maintaining orthogonality of the space-time mesh and pre-integrating analytically through the time-slab, we obtain an efficient yet robust local space-time adaptive method. Any standard spatial element may be used together with standard spatial mesh generation and visualization methods. Recovery based error estimates are used in both space and time dimensions to determine the number and size of local space-time elements within a global time step such that both the spatial and temporal estimated error is equally distributed throughout the space-time approximation. The result is an efficient and reliable adaptive strategy which distributes local space-time elements where needed to accurately track time-dependent waves over large distances and time. Numerical examples of time-dependent acoustic radiation are given which demonstrate the accuracy, reliability and efficiency gained from this new technology.

Adaptive Time-Discontinuous Galerkin Methods for Acoustic Scattering in Unbounded Domains

2002 ◽  
Author(s):  
Lonny L. Thompson ◽  
Dantong He
Keyword(s):  
Discontinuous Galerkin ◽  
Data Transfer ◽  
Hyperbolic Equations ◽  
Acoustic Scattering ◽  
Cost Savings ◽  
Adaptive Strategy ◽  
Space Time ◽  
Time Step ◽  
Solution Algorithms ◽  
Adaptive Procedures

Comprehensive adaptive procedures with efficient solution algorithms for the time-discontinuous Galerkin space-time finite element method (DGFEM) including high-order accurate nonreflecting boundary conditions (NRBC) for unbounded wave problems are developed. Sparse multi-level iterative schemes are developed to solve the resulting system equations for the interior hyperbolic equations coupled with the first-order equations associated with auxiliary functions in the NRBC. The iterative strategy requires only a f ew iterations per time step to resolve the solution to high accuracy. Further cost savings are obtained by diagonalizing the mass and boundary damping matrices. In this case the algebraic structure decouples the diagonal block matrices giving rise to an unconditionally stable explicit iterative method. An h-adaptive space-time strategy is employed based on the superconvergent patch recovery (SPR) technique, together with a temporal error estimate arising from the discontinuous jump between time steps. For accurate data transfer (projection) between meshes, we develop a new superconvergent interpolation (SI) method. Numerical studies of transient scattering demonstrate the accuracy, reliability and efficiency gained from the adaptive strategy.

scholarly journals Some Results for a Time Interval Approach to Field Theory and Gravitation

2021 ◽  
Author(s):  
Harmen Henricus Hollestelle
Keyword(s):  
Field Theory ◽  
Space Time ◽  
Time Dependent ◽  
Field Energy ◽  
Time Interval ◽  
Time Intervals ◽  
Metric Tensor ◽  
Zero Mass ◽  
Interval Approach ◽  
Wave Emission

This paper consists of two parts. In part I some new relations for a field theory with time intervals are derived. One concept of field theory evaluated is complementarity, another is field operators both defined within a time interval description. Part II includes specific results and commentary. Discussed are time interval dependent wave propagation surfaces for star source emission waves and derived is a metric propagation surface area requirement. The results allow to consider one same field that like gravitation within General Relativity applies to both non zero and zero mass. The associated field energy is space time dependent for non zero mass, and is related to a space time dependent metric tensor for zero mass wave particles. Defined is internal energy transfer where wave particle numbers increase linearly and mass and momentum diminish, decrease inversely with the distance from the wave emission source. The commentary are applications related to cosmological overall volume and temperature dependence.

Some Results for a Time Interval Approach to Field Theory and Gravitation

2021 ◽  
Author(s):  
Harmen Henricus Hollestelle
Keyword(s):  
Field Theory ◽  
Space Time ◽  
Time Dependent ◽  
Field Energy ◽  
Time Interval ◽  
Time Intervals ◽  
Metric Tensor ◽  
Zero Mass ◽  
Interval Approach ◽  
Wave Emission

This paper consists of two parts. In part I some new relations for a field theory with time intervals are derived. One concept of field theory evaluated is complementarity, another is operators however both defined within a time interval description. Part II includes specific results and commentary. Discussed are time interval dependent wave propagation surfaces for star source emission waves and derived is a metric surface area requirement for propagation surfaces. The results allow to consider one field that like gravitation within General Relativity applies to both non zero and zero mass. The associated field energy is space time dependent for non zero mass, and is related to a space time dependent metric tensor for zero mass wave particles. Defined is internal energy transfer where wave particle numbers increase linearly and mass and momentum diminish, decrease inversely with the distance from the wave emission source. The commentary are applications related to cosmological overall volume and temperature dependence.

A Compact High Order Space-Time Method for Conservation Laws

2011 ◽  
Vol 9 (2) ◽  
pp. 441-480 ◽  
Author(s):  
Shuangzhang Tu ◽  
Gordon W. Skelton ◽  
Qing Pang
Keyword(s):  
Conservation Laws ◽  
Discontinuous Galerkin ◽  
Hyperbolic Conservation Laws ◽  
Space Time ◽  
High Order ◽  
Basis Functions ◽  
High Order Accuracy ◽  
Time Step ◽  
Order Of Accuracy ◽  
Hyperbolic Conservation

AbstractThis paper presents a novel high-order space-time method for hyperbolic conservation laws. Two important concepts, the staggered space-time mesh of the space-time conservation element/solution element (CE/SE) method and the local discontinuous basis functions of the space-time discontinuous Galerkin (DG) finite element method, are the two key ingredients of the new scheme. The staggered space-time mesh is constructed using the cell-vertex structure of the underlying spatial mesh. The universal definitions of CEs and SEs are independent of the underlying spatial mesh and thus suitable for arbitrarily unstructured meshes. The solution within each physical time step is updated alternately at the cell level and the vertex level. For this solution updating strategy and the DG ingredient, the new scheme here is termed as the discontinuous Galerkin cell-vertex scheme (DG-CVS). The high order of accuracy is achieved by employing high-order Taylor polynomials as the basis functions inside each SE. The present DG-CVS exhibits many advantageous features such as Riemann-solver-free, high-order accuracy, point-implicitness, compactness, and ease of handling boundary conditions. Several numerical tests including the scalar advection equations and compressible Euler equations will demonstrate the performance of the new method.

scholarly journals Space-Time Discontinuous Galerkin Discretizations for Linear First-Order Hyperbolic Evolution Systems

2016 ◽  
Vol 16 (3) ◽  
pp. 409-428 ◽  
Author(s):  
Willy Dörfler ◽  
Stefan Findeisen ◽  
Christian Wieners
Keyword(s):  
Discontinuous Galerkin ◽  
Galerkin Approximation ◽  
Solution Process ◽  
Time Discretization ◽  
Adaptive Strategy ◽  
Space Time ◽  
First Order ◽  
Linear Transport Equation ◽  
Adaptive Solution ◽  
Evolution Systems

AbstractWe introduce a space-time discretization for linear first-order hyperbolic evolution systems using a discontinuous Galerkin approximation in space and a Petrov–Galerkin scheme in time. We show well-posedness and convergence of the discrete system. Then we introduce an adaptive strategy based on goal-oriented dual-weighted error estimation. The full space-time linear system is solved with a parallel multilevel preconditioner. Numerical experiments for the linear transport equation and the Maxwell equation in 2D underline the efficiency of the overall adaptive solution process.

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