Improved Accuracy of High-Order WENO Finite Volume Methods on Cartesian Grids

We propose a simple modification of standard weighted essentially non-oscillatory (WENO) finite volume methods for Cartesian grids, which retains the full spatial order of accuracy of the one-dimensional discretization when applied to nonlinear multidimensional systems of conservation laws. We derive formulas, which allow us to compute high-order accurate point values of the conserved quantities at grid cell interfaces. Using those point values, we can compute a high-order flux at the center of a grid cell interface. Finally, we use those point values to compute high-order accurate averaged fluxes at cell interfaces as needed by a finite volume method. The method is described in detail for the two-dimensional Euler equations of gas dynamics. An extension to the three-dimensional case as well as to other nonlinear systems of conservation laws in divergence form is straightforward. Furthermore, similar ideas can be used to improve the accuracy of WENO type methods for hyperbolic systems which are not in divergence form. Several test computations confirm the high-order accuracy for smooth nonlinear problems.

[1]  Chi-Wang Shu,et al.  A technique of treating negative weights in WENO schemes , 2000 .

[2]  Phillip Colella,et al.  A HIGH-ORDER FINITE-VOLUME METHOD FOR CONSERVATION LAWS ON LOCALLY REFINED GRIDS , 2011 .

[3]  Bertram Taetz,et al.  A high-order unstaggered constrained transport method for the 3D ideal magnetohydrodynamic equations based on the method of lines , 2012, 1203.3760.

[4]  Bertram Taetz,et al.  An unstaggered constrained transport method for the 3D ideal magnetohydrodynamic equations , 2010, J. Comput. Phys..

[5]  Harold L. Atkins,et al.  A Finite-Volume High-Order ENO Scheme for Two-Dimensional Hyperbolic Systems , 1993 .

[6]  BuchmüllerPawel,et al.  Improved Accuracy of High-Order WENO Finite Volume Methods on Cartesian Grids , 2014 .

[7]  James A. Rossmanith,et al.  Finite difference weighted essentially non-oscillatory schemes with constrained transport for 2D ideal Magnetohydrodynamics , 2013, ICOPS 2013.

[8]  Mengping Zhang,et al.  On the Order of Accuracy and Numerical Performance of Two Classes of Finite Volume WENO Schemes , 2011 .

[9]  Chi-Wang Shu,et al.  Monotonicity Preserving Weighted Essentially Non-oscillatory Schemes with Increasingly High Order of Accuracy , 2000 .

[10]  Wai-Sun Don,et al.  Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes , 2013, J. Comput. Phys..

[11]  Klassische Runge-Kutta-Formeln fünfter und siebenter Ordnung mit Schrittweiten-Kontrolle , 2005, Computing.

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

[13]  Randall J. LeVeque,et al.  High-Order Wave Propagation Algorithms for Hyperbolic Systems , 2011, SIAM J. Sci. Comput..

[14]  Qi Tang,et al.  Finite difference weighted essentially non-oscillatory schemes with constrained transport for ideal magnetohydrodynamics , 2013, J. Comput. Phys..

[15]  Barry Merriman,et al.  Understanding the Shu–Osher Conservative Finite Difference Form , 2003, J. Sci. Comput..

[16]  Eleuterio F. Toro,et al.  Finite-volume WENO schemes for three-dimensional conservation laws , 2004 .

[17]  Vladimir A. Titarev,et al.  WENO schemes on arbitrary mixed-element unstructured meshes in three space dimensions , 2011, J. Comput. Phys..

[18]  Fudziah Ismail,et al.  Fifth-Order Improved Runge-Kutta Method With Reduced Number of Function Evaluations , 2012 .

[19]  J. M. Powers,et al.  Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points , 2005 .

[20]  Chaowei Hu,et al.  No . 98-32 Weighted Essentially Non-Oscillatory Schemes on Triangular Meshes , 1998 .

[21]  Gecheng Zha,et al.  Improved Seventh-Order WENO Scheme , 2010 .

[22]  James A. Rossmanith,et al.  An Unstaggered, High-Resolution Constrained Transport Method for Magnetohydrodynamic Flows , 2006, SIAM J. Sci. Comput..

[23]  Bertram Taetz,et al.  A High-Order Unstaggered Constrained-Transport Method for the Three-Dimensional Ideal Magnetohydrodynamic Equations Based on the Method of Lines , 2013, SIAM J. Sci. Comput..

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

[25]  Chaopeng Shen,et al.  Adaptive mesh refinement based on high order finite difference WENO scheme for multi-scale simulations , 2011, J. Comput. Phys..

[26]  C. Schulz-Rinne,et al.  Classification of the Riemann problem for two-dimensional gas dynamics , 1991 .

[27]  Chi-Wang Shu,et al.  High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems , 2009, SIAM Rev..