ENO adaptive method for solving one-dimensional conservation laws

In this work an efficient third order non-linear finite difference scheme for solving adaptively hyperbolic systems of one-dimensional conservation laws is developed. The method is based on applying to the solution of the differential equation an interpolating wavelet transform at each time step, generating a multilevel representation for the solution, which is thresholded and a sparse point representation is generated. The numerical fluxes obtained by a Lax-Friedrichs flux splitting are evaluated on the sparse grid by an essentially non-oscillatory (ENO) approximation, which chooses the locally smoothest stencil among all the possibilities for each point of the sparse grid. The time evolution of the differential operator is done on this sparse representation by a total variation diminishing (TVD) Runge-Kutta method. Four classical examples of initial value problems for the Euler equations of gas dynamics are accurately solved and their sparse solutions are analyzed with respect to the threshold parameters, confirming the efficiency of the wavelet transform as an adaptive grid generation technique.

[1]  Oleg V. Vasilyev,et al.  An Adaptive Wavelet Collocation Method for Fluid-Structure Interaction at High Reynolds Numbers , 2005, SIAM J. Sci. Comput..

[2]  Stanley Osher,et al.  Convex ENO High Order Multi-dimensional Schemes without Field by Field Decomposition or Staggered Grids , 1998 .

[3]  Sônia M. Gomes,et al.  A Fully Adaptive Multiresolution Scheme for Shock Computations , 2001 .

[4]  D. Donoho.  Unconditional Bases Are Optimal Bases for Data Compression and for Statistical Estimation , 1993 .

[5]  Rosa Donat,et al.  Studies on error propagation for certain nonlinear approximations to hyperbolic equations: discontinuities in derivatives , 1994 .

[6]  Chi-Wang Shu,et al.  Total variation diminishing Runge-Kutta schemes , 1998, Math. Comput..

[7]  Olivier Roussel,et al.  A conservative fully adaptive multiresolution algorithm for parabolic PDEs , 2003 .

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

[9]  M. Berger,et al.  Adaptive mesh refinement for hyperbolic partial differential equations , 1982 .

[10]  Stanley Osher,et al.  A second order primitive preconditioner for solving all speed multi-phase flows , 2005 .

[11]  I. Daubechies,et al.  Multiresolution analysis, wavelets and fast algorithms on an interval , 1993 .

[12]  Chi-Wang Shu.  Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws , 1998 .

[13]  Adhemar Bultheel,et al.  Multiple wavelet threshold estimation by generalized cross validation for images with correlated noise , 1999, IEEE Trans. Image Process..

[14]  Sônia M. Gomes,et al.  Adaptive wavelet representation and differentiation on block-structured grids , 2003 .

[15]  Guus Segal,et al.  A Conserving Discretization for the Free Boundary in a Two-Dimensional Stefan Problem , 1998 .

[16]  Samuel Paolucci,et al.  A multilevel wavelet collocation method for solving partial differential equations in a finite domain , 1995 .

[17]  Frédérique Plantevin,et al.  Wavelets on irregular meshes , 1995, Advances in Computational Mathematics.

[18]  Nira Dyn,et al.  Interpolation and Approximation of Piecewise Smooth Functions , 2005, SIAM J. Numer. Anal..

[19]  J. Strikwerda.  Finite Difference Schemes and Partial Differential Equations , 1989 .

[20]  Barna L. Bihari,et al.  Application of generalized wavelets: an adaptive multiresolution scheme , 1995 .

[21]  Mauricio Sepúlveda,et al.  Multiresolution schemes for strongly degenerate parabolic equations in one space dimension , 2007 .

[22]  S. Osher,et al.  Uniformly high order accurate essentially non-oscillatory schemes, 111 , 1987 .

[23]  E. Toro.  Godunov Methods: Theory and Applications , 2001 .

[24]  Rosa Donat,et al.  Point Value Multiscale Algorithms for 2D Compressible Flows , 2001, SIAM J. Sci. Comput..

[25]  Albert Cohen,et al.  Wavelet methods in numerical analysis , 2000 .

[26]  S. Osher,et al.  Very High Order Accurate TVD Schemes , 1986 .

[27]  Mats Holmström,et al.  Solving Hyperbolic PDEs Using Interpolating Wavelets , 1999, SIAM J. Sci. Comput..

[28]  Leland Jameson,et al.  A Wavelet-Optimized, Very High Order Adaptive Grid and Order Numerical Method , 1998, SIAM J. Sci. Comput..

[29]  A. Harten.  Adaptive Multiresolution Schemes for Shock Computations , 1994 .

[30]  Francesc Aràndiga,et al.  Multiresolution Based on Weighted Averages of the Hat Function II: Nonlinear Reconstruction Techniques , 1998, SIAM J. Sci. Comput..

[31]  S. Dubuc.  Interpolation through an iterative scheme , 1986 .

[32]  Björn Sjögreen,et al.  Numerical experiments with the multiresolution scheme for the compressible Euler equations , 1995 .

[33]  Alice J. Kozakevicius,et al.  Adaptive multiresolution WENO schemes for multi-species kinematic flow models , 2007, J. Comput. Phys..

[34]  Leland Jameson,et al.  ON THE WAVELET OPTIMIZED FINITE DIFFERENCE METHOD , 1994 .

[35]  J. Stoer,et al.  Introduction to Numerical Analysis , 2002 .

[36]  A. Harten.  Multiresolution algorithms for the numerical solution of hyperbolic conservation laws , 2010 .