We describe a strategy for detecting discontinuities and for limiting spurious oscillations near such discontinuities when solving hyperbolic systems of conservation laws by high-order discontinuous Galerkin methods. The approach is based on a strong superconvergence at the outflow boundary of each element in smooth regions of the flow. By detecting discontinuities in such variables as density ...

In this paper, we present first, second, and third order implicit finite-volume solvers for advectiondiffusion problems based on the first-order hyperbolic system method. In particular, we demonstrate that the construction of an uniformly accurate third-order advection-diffusion scheme is made trivial by the hyperbolic method while a naive construction of adding a third-order diffusion scheme t...

2 Determining the order of accuracy We are often faced with the problem of how to determine the order p given a sequence of approximations ũh1 , ũh2 , . . . This is can be a good check that a method is correctly implemented (if p is known) and also a way to get a feeling for the credibility of an approximation ũh (high p means high credibility). We can either be in the situation that the exact ...

A new unified theory underlying the theoretical design of linear computational algorithms in the context of time dependent first-order systems is presented. Providing for the first time new perspectives and fresh ideas, and unlike various formulations existing in the literature, the present unified theory involves the following considerations: (i) it leads to new avenues for designing new compu...

Theoretically and computationally, it is possible to demonstrate that the order of accuracy of a discontinuous Galerkin (DG) solution for linear hyperbolic equations can be improved from order k+1 to 2k+1 through the use of smoothness-increasing accuracy-conserving (SIAC) filtering. However, it is a computationally complex task to perform this in an efficient manner, which becomes an even great...

Semi-Lagrangian schemes have been explored by several authors recently for transport problems, in particular for moving interfaces using the level set method. We incorporate the backward error compensation method developed in our paper from 2003 into semi-Lagrangian schemes with almost the same simplicity and three times the complexity of a first order semi-Lagrangian scheme but with improved o...

We study schemes for interpolating functions that take values in the special orthogonal group SO(n). Our focus is on interpolation schemes obtained by embedding SO(n) in a linear space, interpolating in the linear space, and mapping the result onto SO(n) via the closest point projection. The resulting interpolants inherit both the order of accuracy and the regularity of the underlying interpola...

In this paper, electromagnetic wave propagation close to a material discontinuity have been simulated. We have used summation by part (SBP) operators of second, fourth and sixth order accuracy. The boundary conditions have been imposed by the simultaneous approximation term (SAT) procedure. Stability has been shown and the order of accuracy have been verified numerically.

A procedure to locally change the order of accuracy of finite difference schemes is developed. The development is based on existing SummationBy-Parts operators and a weak interface treatment. The resulting scheme is proven to be accurate and stable. Numerical experiments verify the theoretical accuracy for smooth solutions. In addition, shock calculations are performed, using a scheme where the...

The hierarchical reconstruction (HR) [Liu, Shu, Tadmor and Zhang, SINUM ’07] can effectively reduce spurious oscillations without local characteristic decomposition for numerical capturing of discontinuous solutions. However, there are still small remaining overshoots/undershoots in the vicinity of discontinuities. HR with partial neighboring cells [Xu, Liu and Shu, JCP ’09] essentially overcom...