Arbitrarily Accurate Non-Oscillatory Schemes for Nonlinear Scalar Conservation Laws with Source Terms II

Direct-expansion forms of ADER schemes for conservation laws and their verification

“…As to basic derivation, see Ref. [11]. When m = r is taken, the resulting ADER schemes have the rth order of accuracy in both time and space.…”

“…Possibility to increase the convergence rate is to use the WENO interpolation [9,1] with r stencils which satisfies (2r À 1)th order of accuracy for q and f (Eqs. (12) and (44)) instead of rth order (Eqs. (11) and (43)); then, as k ðjÞ q ðqÞ ðj ¼ 0; 1; .…”

“…. In both reconstruction ways, when r stencils are used, corresponding to the accuracy of p i (x), the following orders of accuracy are estimated for g i (x) [11] ENO : g i ðxÞ ¼ f ðx; t n Þ þ OðDx r Þ; ð43Þ…”

“…The ADER approach [18][19][20] is to construct explicit, one-step advection schemes with very high order of accuracy in both space and time on the basis of the solution of GRP (generalized Riemann problem) obtained by use of the solutions of the conventional RP (Riemann problem) and derivative RPs (DRPs). Two methods are possible for ADER schemes to solve general nonlinear conservation laws: methods based on state-series expansion [19] and direct expansion [11]. Recently it has been extended, with high accuracy, to nonlinear conservation laws with source terms [12,13] and diffusion terms [14] and to those in multi-space dimensions [22], and applied to the practical fluid dynamics problems such as the Euler equation system [15].…”

“…Two methods are possible for ADER schemes to solve general nonlinear conservation laws: methods based on state-series expansion [19] and direct expansion [11]. Recently it has been extended, with high accuracy, to nonlinear conservation laws with source terms [12,13] and diffusion terms [14] and to those in multi-space dimensions [22], and applied to the practical fluid dynamics problems such as the Euler equation system [15]. This paper is intended to investigate, as a basic research, if the ADER approach can be the general-purpose algorithm.…”