An Essentially Non-Oscillatory Spectral Deferred Correction Method for Conservation Laws

2016 ◽  
Vol 13 (05) ◽  
pp. 1650027 ◽  
Author(s):  
Samet Y. Kadioglu ◽  
Veli Colak
Keyword(s):  
Conservation Laws ◽  
Fourth Order ◽  
Time Integration ◽  
High Order ◽  
Test Problems ◽  
Order Of Accuracy ◽  
Volume Method ◽  
Spectral Deferred Correction ◽  
Spectral Deferred Corrections ◽  
Eno Scheme

We present a computational method based on the Spectral Deferred Corrections (SDC) time integration technique and the Essentially Non-Oscillatory (ENO) finite volume method for the conservation laws (one-dimensional Euler equations). The SDC technique is used to advance the solutions in time with high-order of accuracy. The ENO method is used to define high-order cell edge quantities that are then used to evaluate numerical fluxes. The coupling of the SDC method with a high-order finite volume method (Piece-wise Parabolic Method (PPM)) for solving the conservation laws is first carried out by Layton et al. in [Layton, A. T. and Minion, M. L. [2004] “Conservative multi-implicit spectral deferred correction methods for reacting gas dynamics,” J. Comput. Phys. 194(2), 697–714]. Issues about this approach have been addressed and some improvements have been added to it in [Kadioglu et al. [2012] “A gas dynamics method based on the spectral deferred corrections (SDC) time integration technique and the piecewise parabolic method (PPM),” Am. J. Comput. Math. 1–4, 303–317]. Here, we investigate the implications when the PPM method is replaced with the well-known ENO method. We note that the SDC-PPM method is fourth-order accurate in time and space. Therefore, we kept the order of accuracy of the ENO procedure as fourth-order in order to be able to make a consistent comparison between the two approaches (SDC-ENO versus SDC-PPM methods). We have tested the new SDC-ENO technique by solving several test problems involving moderate to strong shock waves and smooth/complex flow structures. Our numerical results show that we have numerically achieved the formally fourth-order convergence of the new method for smooth problems. Our numerical results also indicate that the newly proposed technique performs very well providing highly resolved shock discontinuities and fairly good contact solutions. More importantly, the discontinuities in the flow test problems are captured with essentially no-oscillations. We have numerically compared the fourth-order SDC-ENO scheme to the fourth-order SDC-PPM method for the same test problems. The results are similar for most of the test problems except in some cases the SDC-PPM method suffers from minor oscillations compared to SDC-ENO scheme being completely oscillation free.

scholarly journals Analysis of chosen numerical methods for the application in high order interior ballistics simulations

2021 ◽  
Vol 2090 (1) ◽  
pp. 012112
Author(s):  
M Krol
Keyword(s):  
High Order ◽  
Test Problems ◽  
Numerical Accuracy ◽  
Order Of Accuracy ◽  
Computing Power ◽  
Interior Ballistics ◽  
Depth Analysis ◽  
Catastrophic Failures ◽  
Volume Method ◽  
The Mathematical Model

Abstract Considering constant development of the interior ballistics, along with new gun and ammunition designs, the necessity of in-depth analysis of the shot event is continuously increasing. Numerical simulations of interior ballistics problems are useful for optimising new designs or explaining complex issues, regarding performance instabilities and catastrophic failures. With the rise of the computing power, there is a significant urge to drive the numerical errors towards machine zero. This goal demands using methods of high order of accuracy in both space and time. Current methods allow to achieve an arbitrary order of numerical accuracy, thus allowing to shift the focus towards sophistication of the mathematical model of the studied phenomenon. Therefore, in this work, some numerical schemes, in context of finite volume method, are reviewed and studied using well established test problems. The results of the presented analysis are meant to become the basis for future development of a high order numerical scheme for simulation of interior ballistics problems.

An Essentially Nonoscillatory Spectral Deferred Correction Method for Hyperbolic Problems

2016 ◽  
Vol 13 (03) ◽  
pp. 1650017 ◽  
Author(s):  
Samet Y. Kadioglu
Keyword(s):  
Finite Volume Method ◽  
Finite Volume ◽  
Fourth Order ◽  
High Order ◽  
Computational Method ◽  
Hyperbolic Problems ◽  
Discontinuous Solutions ◽  
Order Of Accuracy ◽  
Volume Method ◽  
Essentially Nonoscillatory

We present a computational method based on the spectral deferred corrections (SDC) time integration technique and the essentially nonoscillatory (ENO) finite volume method for hyperbolic problems. The SDC technique is used to advance the solutions in time with high-order of accuracy. The ENO method is used to define high-order cell edge quantities that are then used to evaluate numerical fluxes. The coupling of the SDC method with a high-order finite volume method (piece-wise parabolic method (PPM)) is first carried out by Layton et al. [J. Comput. Phys. 194(2) (2004) 697]. Issues about this approach have been addressed and some improvements have been added to it in Kadioglu et al. [J. Comput. Math. 1(4) (2012) 303]. Here, we investigate the implications when the PPM method is replaced with the well-known ENO method. We note that the SDC-PPM method is fourth-order accurate in time and space. Therefore, we kept the order of accuracy of the ENO procedure as fourth-order in order to be able to make a consistent comparison between the two approaches (SDC-ENO versus SDC-PPM). We have tested the new SDC-ENO technique by solving smooth and nonsmooth hyperbolic problems. Our numerical results indicate that the fourth-order of accuracy in both space and time has been achieved for smooth problems. On the other hand, the new method performs very well when it is applied to nonlinear problems that involve discontinuous solutions. In other words, we have obtained highly resolved discontinuous solutions with essentially no-oscillations at or around the discontinuities.

scholarly journals A Slope Constrained 4th Order Multi-Moment Finite Volume Method with WENO Limiter

2015 ◽  
Vol 18 (4) ◽  
pp. 901-930 ◽  
Author(s):  
Ziyao Sun ◽  
Honghui Teng ◽  
Feng Xiao
Keyword(s):  
Conservation Laws ◽  
Finite Volume ◽  
Solution Structure ◽  
Hyperbolic Conservation Laws ◽  
High Order ◽  
Weno Reconstruction ◽  
Hyperbolic Conservation ◽  
Average Value ◽  
High Order Schemes ◽  
Volume Method

AbstractThis paper presents a new and better suited formulation to implement the limiting projection to high-order schemes that make use of high-order local reconstructions for hyperbolic conservation laws. The scheme, so-called MCV-WENO4 (multi-moment Constrained finite Volume with WENO limiter of 4th order) method, is an extension of the MCV method of Ii & Xiao (2009) by adding the 1st order derivative (gradient or slope) at the cell center as an additional constraint for the cell-wise local reconstruction. The gradient is computed from a limiting projection using the WENO (weighted essentially non-oscillatory) reconstruction that is built from the nodal values at 5 solution points within 3 neighboring cells. Different from other existing methods where only the cell-average value is used in the WENO reconstruction, the present method takes account of the solution structure within each mesh cell, and thus minimizes the stencil for reconstruction. The resulting scheme has 4th-order accuracy and is of significant advantage in algorithmic simplicity and computational efficiency. Numerical results of one and two dimensional benchmark tests for scalar and Euler conservation laws are shown to verify the accuracy and oscillation-less property of the scheme.

scholarly journals High Order of Accuracy for Poisson Equation Obtained by Grouping of Repeated Richardson Extrapolation with Fourth Order Schemes

2021 ◽  
Vol 128 (2) ◽  
pp. 699-715
Author(s):  
Luciano Pereira da Silva ◽  
Bruno Benato Rutyna ◽  
Aline Roberta Santos Righi ◽  
Marcio Augusto Villela Pinto
Keyword(s):  
Poisson Equation ◽  
Fourth Order ◽  
Richardson Extrapolation ◽  
High Order ◽  
Order Of Accuracy ◽  
High Order Of Accuracy

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.

Fully and Semi-discrete Fourth-order Schemes for Hyperbolic Conservation Laws

2007 ◽  
Vol 7 (3) ◽  
pp. 264-282
Author(s):  
Y.H. Zahran
Keyword(s):  
Conservation Laws ◽  
Fourth Order ◽  
Hyperbolic Conservation Laws ◽  
Tvd Scheme ◽  
Test Problems ◽  
Hyperbolic Conservation ◽  
Runge Kutta Method ◽  
Order Scheme ◽  
Spurious Oscillations ◽  
Fifth Order

AbstractA new fourth order accurate centered finite difference scheme for the solution of hyperbolic conservation laws is presented. A technique of making the fourth order scheme TVD is presented. The resulting scheme can avoid spurious oscillations and preserve fourth order accuracy in smooth parts. We discuss the extension of the TVD scheme to the nonlinear scalar hyperbolic conservation laws. For nonlinear systems, the TVD constraint is applied by solving shallow water equations. Then, we propose to use this fourth order flux as a building block in spatially fifth order weighted essentially non-oscillatory (WENO) schemes. The numerical solution is advanced in time by the third order TVD Runge — Kutta method. The performance of the scheme is assessed by solving test problems. The numerical results are presented and compared to the exact solutions and other methods.

Sign in / Sign up

Export Citation Format

Share Document