Solutions of Multi-dimensional Hyperbolic Systems of Conservation Laws by Square Entropy Condition Satisfying Discontinuous Galerkin Method

In this paper, we study formally high-order accurate discontinuous Galerkin methods on general arbitrary grid for multi-dimensional hyperbolic systems of conservation laws [Cockburn, B., and Shu, C.-W. (1989, Math. Comput. 52, 411–435, 1998, J. Comput. Phys. 141, 199–224); Cockburn et al. (1989, J. Comput. Phys. 84, 90–113; 1990, Math. Comput. 54, 545–581). We extend the notion of E-flux [Osher (1985) SIAM J. Numer. Anal. 22, 947–961] from scalar to system, and found that after flux splitting upwind flux [Cockburn et al. (1989) J. Comput. Phys. 84, 90–113] is a Riemann solver free E-flux for systems. Therefore, we are able to show that the discontinuous Galerkin methods satisfy a cell entropy inequality for square entropy (in semidiscrete sense) if the multi-dimensional systems are symmetric. Similar result [Jiang and Shu (1994) Math. Comput. 62, 531–538] was obtained for scalar equations in multi-dimensions. We also developed a second-order finite difference version of the discontinuous Galerkin methods. Numerical experiments have been obtained with excellent results.      

Locally Divergence-Free Discontinuous Galerkin Methods for MHD Equations

In this paper, we continue our investigation of the locally divergence-free discontinuous Galerkin method, originally developed for the linear Maxwell equations (J. Comput. Phys. 194 588–610 (2004)), to solve the nonlinear ideal magnetohydrodynamics (MHD) equations. The distinctive feature of such method is the use of approximate solutions that are exactly divergence-free inside each element for the magnetic field. As a consequence, this method has a smaller computational cost than the traditional discontinuous Galerkin method with standard piecewise polynomial spaces. We formulate the locally divergence-free discontinuous Galerkin method for the MHD equations and perform extensive one and two-dimensional numerical experiments for both smooth solutions and solutions with discontinuities. Our computational results demonstrate that the locally divergence-free discontinuous Galerkin method, with a reduced cost comparing to the traditional discontinuous Galerkin method, can maintain the same accuracy for smooth solutions and can enhance the numerical stability of the scheme and reduce certain nonphysical features in some of the test cases.This revised version was published online in July 2005 with corrected volume and issue numbers.     

An analysis and comparison of conservative upwind difference schemes for ideal and non-ideal gases

In a recent paper [P. Glaister, Conservative upwind difference schemes for the Euler equations, Comput. Math. Appl. 45 (2003) 1673–1682] a number of numerical schemes were presented for the Euler equations governing compressible flows of an ideal gas, the principal one of which is based on a conservative linearisation approach. This scheme was subsequently extended to encompass compressible flows of real gases where the equation of state allows for non-ideal gases [P. Glaister, Conservative upwind difference schemes for compressible flows of a real gas, Comput. Math. Appl. 48 (2004) 469–480]. These schemes use different parameter vectors in their construction and, consequently, the scheme in [P. Glaister, Conservative upwind difference schemes for compressible flows of a real gas, Comput. Math. Appl. 48 (2004) 469–480] when applied to the special case of an ideal gas is not identical to the principal ideal gas scheme in [P. Glaister, Conservative upwind difference schemes for the Euler equations, Comput. Math. Appl. 45 (2003) 1673–1682]. In this paper it is shown how these schemes are related, followed by a numerical comparison when each is applied to two standard test problems.     

ADER schemes on unstructured meshes for nonconservative hyperbolic systems: Applications to geophysical flows

We develop a new family of well-balanced path-conservative quadrature-free one-step ADER finite volume and discontinuous Galerkin finite element schemes on unstructured meshes for the solution of hyperbolic partial differential equations with non-conservative products and stiff source terms. The fully discrete formulation is derived using the recently developed framework of explicit one-step P_NP_M schemes of arbitrary high order of accuracy in space and time for conservative hyperbolic systems [Dumbser M, Balsara D, Toro EF, Munz CD. A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes. J Comput Phys 2008;227:8209–53]. The two key ingredients of our high order approach are: first, the high order accurate P_NP_M reconstruction operator on unstructured meshes, using the WENO strategy presented in [Dumbser M, Käser M, Titarev VA Toro EF. Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. J Comput Phys 2007;226:204–43] to ensure monotonicity at discontinuities, and second, a local space–time Galerkin scheme to predict the evolution of the reconstructed polynomial data inside each element during one time step to obtain a high order accurate one-step time discretization. This approach is also able to deal with stiff source terms as shown in [Dumbser M, Enaux C, Toro EF. Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. J Comput Phys 2008;227:3971–4001]. These two key ingredients are combined with the recently developed path-conservative methods of Parés [Parés C. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J Numer Anal 2006;44:300–21] and Castro et al. [Castro MJ, Gallardo JM, Parés C. High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems. Math Comput 2006;75:1103–34] to treat the non-conservative products properly. We show applications of our method to the two-layer shallow water equations as well as applications to the recently published depth-averaged two-fluid flow model of Pitman and Le [Pitman EB, Le L. A two-fluid model for avalanche and debris flows. Philos Trans Roy Soc A 2005;363:1573–601].     

Soft -ideals of soft BCI-algebras

Molodtsov [D. Molodtsov, Soft set theory–First results, Comput. Math. Appl. 37 (1999) 19–31] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Jun [Y. B. Jun, Soft BCK/BCI-algebras, Comput. Math. Appl. 56 (2008) 1408–1413] applied first the notion of soft sets by Molodtsov to the theory of BCK/BCI-algebras. In this paper we introduce the notion of soft p-ideals and p-idealistic soft BCI-algebras, and then investigate their basic properties. Using soft sets, we give characterizations of (fuzzy) p-ideals in BCI-algebras. We provide relations between fuzzy p-ideals and p-idealistic soft BCI-algebras.     

Mixed Discontinuous Galerkin Methods for Darcy Flow

We consider a family of mixed finite element discretizations of the Darcy flow equations using totally discontinuous elements (both for the pressure and the flux variable). Instead of using a jump stabilization as it is usually done for discontinuos Galerkin (DG) methods (see e.g. D.N. Arnold et al. SIAM J. Numer. Anal.39, 1749–1779 (2002) and B. Cockburn, G.E. Karniadakis and C.-W. Shu, DG methods: Theory, computation and applications, (Springer, Berlin, 2000) and the references therein) we use the stabilization introduced in A. Masud and T.J.R. Hughes, Meth. Appl. Mech. Eng.191, 4341–4370 (2002) and T.J.R. Hughes, A. Masud, and J. Wan, (in preparation). We show that such stabilization works for discontinuous elements as well, provided both the pressure and the flux are approximated by local polynomials of degree ≥ 1, without any need for additional jump terms. Surprisingly enough, after the elimination of the flux variable, the stabilization of A. Masud and T.J.R. Hughes, Meth. Appl. Mech. Eng.191, 4341–4370 (2002) and T.J.R. Hughes, A. Masud, and J. Wan, (in preparation) turns out to be in some cases a sort of jump stabilization itself, and in other cases a stable combination of two originally unstable DG methods (namely, Bassi-Rebay F. Bassi and S. Rebay, Proceedings of the Conference ‘‘Numerical methods for fluid dynamics V’‘, Clarendon Press, Oxford (1995) and Baumann–Oden Comput. Meth. Appl. Mech. Eng.175, 311–341 (1999).This revised version was published online in July 2005 with corrected volume and issue numbers.     

L2(H1 Norm A PosterioriError Estimation for Discontinuous Galerkin Approximations of Reactive Transport Problems

Explicita posteriori residual type error estimators in L²(H¹) norm are derived for discontinuous Galerkin (DG) methods applied to transport in porous media with general kinetic reactions. They are flexible and apply to all the four primal DG schemes, namely, Oden–Babuška–Baumann DG (OBB-DG), non-symmetric interior penalty Galerkin (NIPG), symmetric interior penalty Galerkin (SIPG) and incomplete interior penalty Galerkin (IIPG). The error estimators use directly all the available information from the numerical solution and can be computed efficiently. Numerical examples are presented to demonstrate the efficiency and the effectivity of these theoretical estimators.The authors would like to thank the anonymous referees for their incisive suggestions which contributed toward improving the paper. AMS subject classifications:65M15;65M60;65M50.This revised version was published online in July 2005 with corrected volume and issue numbers.     

Jungck-Khan iterative scheme and higher convergence rate

In this paper, we continue the theme of analytical and numerical treatment of Jungck-type iterative schemes. In particular, we focus on a special case of Jungck-Khan iterative scheme introduced by Khan et al. [Analytical and numerical treatment of Jungck-type iterative schemes, Appl. Math. Comput. 231 (2014) 521–535] to get an insight in the strong convergence and data dependence results obtained therein. Our investigations show that this special case under different control conditions on parametric sequences provides higher convergence rate and better data dependence estimates as compared to the Jungck-Khan iterative scheme itself.     

One-sided Post-processing for the Discontinuous Galerkin Method Using ENO Type Stencil Choosing and the Local Edge Detection Method

In a previous paper by Ryan and Shu [Ryan, J. K., and Shu, C.-W. (2003). \hboxMethods Appl. Anal. 10(2), 295–307], a one-sided post-processing technique for the discontinuous Galerkin method was introduced for reconstructing solutions near computational boundaries and discontinuities in the boundaries, as well as for changes in mesh size. This technique requires prior knowledge of the discontinuity location in order to determine whether to use centered, partially one-sided, or one-sided post-processing. We now present two alternative stencil choosing schemes to automate the choice of post-processing stencil. The first is an ENO type stencil choosing procedure, which is designed to choose centered post-processing in smooth regions and one-sided or partially one-sided post-processing near a discontinuity, and the second method is based on the edge detection method designed by Archibald, Gelb, and Yoon [Archibald, R., Gelb, A., and Yoon, J. (2005). SIAM J. Numeric. Anal. 43, 259–279; Archibald, R., Gelb, A., and Yoon, J. (2006). Appl. Numeric. Math. (submitted)]. We compare these stencil choosing techniques and analyze their respective strengths and weaknesses. Finally, the automated stencil choices are applied in conjunction with the appropriate post-processing procedures and it is determine that the resulting numerical solutions are of the correct order.     

A Numerical Comparison of the Lax–Wendroff Discontinuous Galerkin Method Based on Different Numerical Fluxes

Discontinuous Galerkin (DG) method is a spatial discretization procedure, employing useful features from high-resolution finite volume schemes, such as the exact or approximate Riemann solvers serving as numerical fluxes and limiters. In [(2005). Comput. Methods Appl. Mech. Eng. 194, 4528], we developed a Lax–Wendroff time discretization procedure for the DG method (LWDG), an alternative method for time discretization to the popular total variation diminishing (TVD) Runge–Kutta time discretizations. In most of the DG papers in the literature, the Lax–Friedrichs numerical flux is used due to its simplicity, although there are many other numerical fluxes, which could also be used. In this paper, we systematically investigate the performance of the LWDG method based on different numerical fluxes, including the first-order monotone fluxes such as the Godunov flux, the Engquist–Osher flux, etc., the second-order TVD fluxes and generalized Riemann solver, with the objective of obtaining better performance by choosing suitable numerical fluxes. The detailed numerical study is mainly performed for the one-dimensional system case, addressing the issues of CPU cost, accuracy, non-oscillatory property, and resolution of discontinuities. Numerical tests are also performed for two-dimensional systems.      

A New Smoothness Indicator for the WENO Schemes and Its Effect on the Convergence to Steady State Solutions

The convergence to steady state solutions of the Euler equations for the fifth-order weighted essentially non-oscillatory (WENO) finite difference scheme with the Lax–Friedrichs flux splitting [7, (1996) J. Comput. Phys. 126, 202–228.] is studied through systematic numerical tests. Numerical evidence indicates that this type of WENO scheme suffers from slight post-shock oscillations. Even though these oscillations are small in magnitude and do not affect the “essentially non-oscillatory” property of WENO schemes, they are indeed responsible for the numerical residue to hang at the truncation error level of the scheme instead of settling down to machine zero. We propose a new smoothness indicator for the WENO schemes in steady state calculations, which performs better near the steady shock region than the original smoothness indicator in [7, (1996) J. Comput. Phys. 126, 202–228.]. With our new smoothness indicator, the slight post-shock oscillations are either removed or significantly reduced and convergence is improved significantly. Numerical experiments show that the residue for the WENO scheme with this new smoothness indicator can converge to machine zero for one and two dimensional (2D) steady problems with strong shock waves when there are no shocks passing through the domain boundaries. Dedicated to the memory of Professor Xu-Dong Liu.     

An Antidiffusive Entropy Scheme for Monotone Scalar Conservation Laws

In a recent work J. Sci. Comput. 16, 479–524 (2001), B. Després and F. Lagoutière introduced a new approach to derive numerical schemes for hyperbolic conservation laws. Its most important feature is the ability to perform an exact resolution for a single traveling discontinuity. However their scheme is not entropy satisfying and can keep nonentropic discontinuities. The purpose of our work is, starting from the previous one, to introduce a new class of schemes for monotone scalar conservation laws, that satisfy an entropy inequality, while still resolving exactly the single traveling shocks or contact discontinuities. We show that it is then possible to have an excellent resolution of rarefaction waves, and also to avoid the undesirable staircase effect. In practice, our numerical experiments show second-order accuracy.     

Spline collocation method with four parameters for solving a system of fourth-order boundary-value problems

In this paper, a spline collocation method is applied to solve a system of fourth-order boundary-value problems associated with obstacle, unilateral and contact problems. The presented method is dependent on four collocation points to be satisfied by four parameters θ _j ∈(0, 1], j=1(1) 4 in each subinterval. It turns out that the proposed method when applied to the concerned system is a fourth-order convergent method and gives numerical results which are better than those produced by other spline methods [E.A. Al-Said and M.A. Noor, Finite difference method for solving fourth-order obstacle problems, Int. J. Comput. Math. 81(6) (2004), pp. 741–748; F. Geng and Y. Lin, Numerical solution of a system of fourth order boundary value problems using variational iteration method, Appl. Math. Comput. 200 (2008), pp. 231–241; J. Rashidinia, R. Mohammadi, R. Jalilian, and M. Ghasemi, Convergence of cubic-spline approach to the solution of a system of boundary-value problems, Appl. Math. Comput. 192 (2007), pp. 319–331; S.S. Siddiqi and G. Akram, Solution of the system of fourth order boundary value problems using non polynomial spline technique, Appl. Math. Comput. 185 (2007), pp. 128–135; S.S. Siddiqi and G. Akram, Numerical solution of a system of fourth order boundary value problems using cubic non-polynomial spline method, Appl. Math. Comput. 190(1) (2007), pp. 652–661; S.S. Siddiqi and G. Akram, Solution of the system of fourth order boundary value problems using cubic spline, Appl. Math. Comput. 187(2) (2007), pp. 1219–1227; Siraj-ul-Islam, I.A. Tirmizi, F. Haq, and S.K. Taseer, Family of numerical methods based on non-polynomial splines for solution of contact problems, Commun. Nonlinear Sci. Numer. Simul. 13 (2008), pp. 1448–1460]. Moreover, the absolute stability properties appear that the method is A-stable. Two numerical examples (one for each case of boundary conditions) are given to illustrate practical usefulness of the method developed.     

A discontinuous finite difference streamline diffusion method for time-dependent hyperbolic problems

In this article, a new finite element method, discontinuous finite difference streamline diffusion method (DFDSD), is constructed and studied for first-order linear hyperbolic problems. This method combines the benefit of the discontinuous Galerkin method and the streamline diffusion finite element method. Two fully discrete DFDSD schemes (Euler DFDSD and Crank–Nicolson (CN) DFDSD) are constructed by making use of the difference discrete method for time variables and the discontinuous streamline diffusion method for space variables. The stability and optimal L² norm error estimates are established for the constructed schemes. This method makes contributions to the discontinuous methods. Finally, a numerical example is provided to show the benefit of high efficiency and simple implementation of the schemes.     

A fast numerical algorithm for solving nearly penta-diagonal linear systems

In this paper, we present a fast numerical algorithm for solving nearly penta-diagonal linear systems and show that the computational cost is less than those of three algorithms in El-Mikkawy and Rahmo, [Symbolic algorithm for inverting cyclic penta-diagonal matrices recursively–Derivation and implementation, Comput. Math. Appl. 59 (2010), pp. 1386–1396], Lv and Le [A note on solving nearly penta-diagonal linear systems, Appl. Math. Comput. 204 (2008), pp. 707–712] and Neossi Nguetchue and Abelman [A computational algorithm for solving nearly penta-diagonal linear systems, Appl. Math. Comput. 203 (2008), pp. 629–634.]. In addition, an efficient way of evaluating the determinant of a nearly penta-diagonal matrix is also discussed. The algorithm is suited for implementation using computer algebra systems (CAS) such as MATLAB, MACSYMA and MAPLE. Some numerical examples are given in order to illustrate the efficiency of our algorithm.     

A variable-mesh approximation method for singularly perturbed boundary-value problems using cubic spline in tension

A non-uniform mesh difference scheme using cubic spline in tension is presented to solve a class of non-turning point singularly perturbed two point boundary-value problems for second-order ordinary differential equations with a small parameter multiplying the highest derivative subject to Dirichlet-type boundary conditions. To demonstrate the applicability of the proposed method, two numerical examples have been solved and the results are presented along with their comparison with those obtained with and without variable mesh. This paper is a continuation of the previous work [Aziz, T. and Khan, A. (2002). A spline method for second order singularly-perturbed boundary-value problems. J. Comput. Appl. Math., 147(2), 445–452.] given for uniform mesh case.     

Stabilized discontinuous Galerkin method for hyperbolic equations

In this work a new stabilization technique is proposed and studied for the discontinuous Galerkin method applied to hyperbolic equations. In order to avoid the use of slope limiters, a streamline diffusion-like term is added to control oscillations for arbitrary element orders. Thus, the scheme combines ideas from both the Runge-Kutta discontinuous Galerkin method [J. Scient. Comput. 16 (2001) 173] and the streamline diffusion method [Comput. Methods Appl. Mech. Engrg. 32 (1982)]. To increase the stability range of the method, the diffusion term is treated implicitly. The result is a scheme with higher order in space with the same stability range as the finite volume method. An optimal relation between the time step and the size of the diffusion coefficient is analyzed for numerical precision. The scheme is implemented using the object oriented programming philosophy based on the environment described in [Comput. Methods Appl. Mech. Engrg. 150 (1997)]. Accuracy and shock capturing abilities of the method are analyzed in terms of two bidimensional model problems: the rotating cone and the backward facing step problem for the Euler equations of gas dynamics.     

Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems

In this paper, we review the development of the Runge–Kutta discontinuous Galerkin (RKDG) methods for non-linear convection-dominated problems. These robust and accurate methods have made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. They combine a special class of Runge–Kutta time discretizations, that allows the method to be non-linearly stable regardless of its accuracy, with a finite element space discretization by discontinuous approximations, that incorporates the ideas of numerical fluxes and slope limiters coined during the remarkable development of the high-resolution finite difference and finite volume schemes. The resulting RKDG methods are stable, high-order accurate, and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions. We review the theoretical and algorithmic aspects of these methods and show several applications including nonlinear conservation laws, the compressible and incompressible Navier–Stokes equations, and Hamilton–Jacobi-like equations.     

A discontinuous Galerkin solver for Boltzmann–Poisson systems in nano devices

In this paper, we present results of a discontinuous Galerkin (DG) scheme applied to deterministic computations of the transients for the Boltzmann–Poisson system describing electron transport in semiconductor devices. The collisional term models optical-phonon interactions which become dominant under strong energetic conditions corresponding to nano-scale active regions under applied bias. The proposed numerical technique is a finite element method using discontinuous piecewise polynomials as basis functions on unstructured meshes. It is applied to simulate hot electron transport in bulk silicon, in a silicon n⁺–n–n⁺ diode and in a double gated 12 nm MOSFET. Additionally, the obtained results are compared to those of a high order WENO scheme simulation and DSMC (Discrete Simulation Monte Carlo) solvers.     

An estimate on the convergence of MKZ–Bézier operators

The pointwise approximation properties of the MKZ–Bézier operators M_n,α(f,x) for α≥1 have been studied in [X.M. Zeng, Rates of approximation of bounded variation functions by two generalized Meyer–König–Zeller type operators, Comput. Math. Appl. 39 (2000) 1–13]. The aim of this paper is to study the pointwise approximation of the operators M_n,α(f,x) for the other case 0<α<1. By means of some new estimate techniques and a result of Guo and Qi [S. Guo, Q. Qi, The moments for the Meyer–König and Zeller operators, Appl. Math. Lett. 20 (2007) 719–722], we establish an estimate formula on the rate of convergence of the operators M_n,α(f,x) for the case 0<α<1.