Sparse Grid Central Discontinuous Galerkin Method for Linear Hyperbolic Systems in High Dimensions

In this paper, we develop sparse grid central discontinuous Galerkin (CDG) scheme for linear hyperbolic systems with variable coefficients in high dimensions. The scheme combines the CDG framework with the sparse grid approach, with the aim of breaking the curse of dimensionality. A new hierarchical representation of piecewise polynomials on the dual mesh is introduced and analyzed, resulting in a sparse finite element space that can be used for non-periodic problems. Theoretical results, such as $L^2$ stability and error estimates are obtained for scalar problems. CFL conditions are studied numerically comparing discontinuous Galerkin (DG), CDG, sparse grid DG and sparse grid CDG methods. Numerical results including scalar linear equations, acoustic and elastic waves are provided.

[1]  Richard Bellman,et al.  Adaptive Control Processes: A Guided Tour , 1961, The Mathematical Gazette.

[2]  M. Dumbser,et al.  An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes — II. The three-dimensional isotropic case , 2006 .

[3]  E. M. Wright,et al.  Adaptive Control Processes: A Guided Tour , 1961, The Mathematical Gazette.

[4]  E. Tadmor,et al.  New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection—Diffusion Equations , 2000 .

[5]  Michael Griebel,et al.  Adaptive sparse grid multilevel methods for elliptic PDEs based on finite differences , 1998, Computing.

[6]  Xiang Ma,et al.  An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations , 2009, J. Comput. Phys..

[7]  Chi-Wang Shu,et al.  Central Discontinuous Galerkin Methods on Overlapping Cells with a Nonoscillatory Hierarchical Reconstruction , 2007, SIAM J. Numer. Anal..

[8]  R. LeVeque Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[9]  Jie Shen,et al.  Efficient Spectral Sparse Grid Methods and Applications to High-Dimensional Elliptic Problems , 2010, SIAM J. Sci. Comput..

[10]  M. Dumbser,et al.  An arbitrary high-order Discontinuous Galerkin method for elastic waves on unstructured meshes — III. Viscoelastic attenuation , 2007 .

[11]  Jie Shen,et al.  A NODAL SPARSE GRID SPECTRAL ELEMENT METHOD FOR MULTI-DIMENSIONAL ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS , 2017 .

[12]  Wei Guo,et al.  Sparse grid discontinuous Galerkin methods for high-dimensional elliptic equations , 2015, J. Comput. Phys..

[13]  Jie Shen,et al.  Sparse Spectral Approximations of High-Dimensional Problems Based on Hyperbolic Cross , 2010, SIAM J. Numer. Anal..

[14]  Fabio Nobile,et al.  A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data , 2008, SIAM J. Numer. Anal..

[15]  Chi-Wang Shu,et al.  Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems , 2001, J. Sci. Comput..

[16]  Michael Griebel,et al.  Adaptive Sparse Grids for Hyperbolic Conservation Laws , 1999 .

[17]  E. Tadmor,et al.  Non-oscillatory central differencing for hyperbolic conservation laws , 1990 .

[18]  Pieter W. Hemker Sparse-grid finite-volume multigrid for 3D-problems , 1995, Adv. Comput. Math..

[19]  M. Dumbser,et al.  An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes - I. The two-dimensional isotropic case with external source terms , 2006 .

[20]  Endre Süli,et al.  Sparse finite element approximation of high-dimensional transport-dominated diffusion problems , 2008 .

[21]  B. Alpert A class of bases in L 2 for the sparse representations of integral operators , 1993 .

[22]  Yingjie Liu Central schemes on overlapping cells , 2005 .

[23]  D. Xiu Efficient collocational approach for parametric uncertainty analysis , 2007 .

[24]  Chi-Wang Shu,et al.  L2 Stability Analysis of the Central Discontinuous Galerkin Method and a Comparison between the Central and Regular Discontinuous Galerkin Methods , 2008 .

[25]  Michael Dumbser,et al.  An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes - IV. Anisotropy , 2007 .

[26]  Michael Griebel,et al.  Sparse Grids and Applications , 2012 .

[27]  Vasile Gradinaru,et al.  Fourier transform on sparse grids: Code design and the time dependent Schrödinger equation , 2007, Computing.

[28]  Wei Guo,et al.  An Adaptive Multiresolution Discontinuous Galerkin Method for Time-Dependent Transport Equations in Multidimensions , 2017, SIAM J. Sci. Comput..

[29]  Jie Shen,et al.  Efficient Spectral Sparse Grid Methods and Applications to High-Dimensional Elliptic Equations II. Unbounded Domains , 2012, SIAM J. Sci. Comput..

[30]  Chi-Wang Shu,et al.  Optimal Error Estimates of the Semidiscrete Central Discontinuous Galerkin Methods for Linear Hyperbolic Equations , 2018, SIAM J. Numer. Anal..

[31]  Fengyan Li,et al.  Operator Bounds and Time Step Conditions for the DG and Central DG Methods , 2015, J. Sci. Comput..

[32]  H. Bungartz,et al.  Sparse grids , 2004, Acta Numerica.

[33]  M. Griebel,et al.  Sparse grids for the Schrödinger equation , 2007 .

[34]  Wei Guo,et al.  A Sparse Grid Discontinuous Galerkin Method for High-Dimensional Transport Equations and Its Application to Kinetic Simulations , 2016, SIAM J. Sci. Comput..

[35]  Yingjie Liu Central Schemes and Central Discontinuous Galerkin Methods on Overlapping Cells , 2005 .

[36]  George Em Karniadakis,et al.  The Development of Discontinuous Galerkin Methods , 2000 .

[37]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[38]  Dongbin Xiu,et al.  High-Order Collocation Methods for Differential Equations with Random Inputs , 2005, SIAM J. Sci. Comput..