Uniformly high order accurate essentially non-oscillatory schemes, 111
Journal of Computational Physics
Efficient implementation of essentially non-oscillatory shock-capturing schemes
Journal of Computational Physics
Journal of Computational Physics
Non-oscillatory central differencing for hyperbolic conservation laws
Journal of Computational Physics
Weighted essentially non-oscillatory schemes
Journal of Computational Physics
Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method
Journal of Computational Physics - Special issue: commenoration of the 30th anniversary
The Runge-Kutta discontinuous Galerkin method for conservation laws V multidimensional systems
Journal of Computational Physics
Nonoscillatory Central Schemes for Multidimensional Hyperbolic Conservation Laws
SIAM Journal on Scientific Computing
SIAM Journal on Numerical Analysis
High-Order Central Schemes for Hyperbolic Systems of Conservation Laws
SIAM Journal on Scientific Computing
Journal of Computational Physics
Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
Journal of Scientific Computing
Semidiscrete Central-Upwind Schemes for Hyperbolic Conservation Laws and Hamilton--Jacobi Equations
SIAM Journal on Scientific Computing
Central schemes on overlapping cells
Journal of Computational Physics
Journal of Computational Physics
SIAM Journal on Numerical Analysis
Efficient implementation of essentially non-oscillatory shock-capturing schemes, II
Journal of Computational Physics
Alternating evolution discontinuous Galerkin methods for Hamilton-Jacobi equations
Journal of Computational Physics
Hi-index | 0.01 |
The alternating evolution (AE) system of Liu [J. Hyperbolic Differ. Equ., 5 (2008), pp. 421-447], $\partial_t u+ \partial_x f(v) =\frac{1}{\epsilon}(v-u)$, $\partial_t v + \partial_x f(u) =\frac{1}{\epsilon}(u-v)$, serves as a refined description of systems of hyperbolic conservation laws $\partial_t \phi+ \partial_x f(\phi)=0$, $\phi(x, 0)=\phi_0(x)$. The solution of conservation laws is precisely captured when two components take the same initial value as $\phi_0$, or is approached by two components exponentially fast when $\epsilon \downarrow 0$ if two initial states are sufficiently close. This nice property enables us to construct novel shock capturing schemes by sampling the AE system on alternating grids. In this paper we develop a class of local AE schemes by taking advantage of the AE system. Our approach is based on an average of the AE system over a hypercube centered at $x$ with vertices at $x\pm\Delta x$. The numerical scheme is then constructed by sampling the averaged system over alternating grids. Higher order accuracy is achieved by a combination of high order nonoscillatory polynomial reconstruction from the obtained averages and a matching Runge-Kutta solver in time discretization. Local AE schemes are made possible by letting the scale parameter $\epsilon$ reflect the local distribution of nonlinear waves. The AE schemes have the advantage of easier formulation and implementation, and efficient computation of the solution. For the first and second order local AE schemes applied to one-dimensional scalar conservation laws, we prove the numerical stability in the sense of satisfying the maximum principle and total variation diminishing property. The formulation procedure of AE schemes in multiple dimensions is given, followed by both the first and second order AE schemes for two-dimensional conservation laws. Numerical experiments for both scalar conservation laws and compressible Euler equations are presented to demonstrate the high order accuracy and capacity of these AE schemes.