Error estimators for the position of discontinuities in hyperbolic conservation laws with source terms which are solved using operator splitting

Abstract

When computing numerical solutions of hyperbolic conservation laws with source terms, one may obtain spurious solutions | these are unphysical solutions that only occur in numerics such a s s h o c k w aves moving with wrong speeds, cf. 7], 2], 1], 10], 3]. Therefore it is important to know h o w errors of the location of a discontinuity can be controlled.To d e r i v e appropriate error-estimates and to use them to control such errors, is the aim of our investigations in this paper. We restrict our considerations t o numerical solutions which are computed by using a splitting method. In splitting methods, the homogeneous conservation law and an ordinary di erential equation (modelling the source term) are solved separately in each t i m e s t e p .Firstly, w e d e r i v e error-estimates for the scalar Riemann problem. The analysis shows that the local error of the location of a discontinuity mainly consists of two parts. The rst part is introduced by the splitting and the second part is due to smearing of the discontinuity.Next, these error-estimates are used to construct an adaptation of the step size so that the error of the location of the discontinuity remains su ciently small. The adaptation is applied to several examples, which are a scalar problem, a simpli ed combustion model, and the one-dimensional i n viscid reacting compressible Euler equations. All the examples show that the adaptation based on the derived error-estimates works well.The theory can also be extended to planar two-dimensional problems.

Numerical solution of under-resolved detonations

“…Eq. (1) is an advection equation with constant propagation speed a and with a nonlinear source term SðuÞ that becomes stiff for large m. The introduction of the more complex advection equation with variable propagation speed, as proposed in [7,17,19], does not modify the mechanism of generation of spurious wave speeds here investigated. Given the selected initial condition, it is easy to verify that the exact solution is…”

“…The scalar model problem, although insufficient to reproduce the physics of rapid combustion, does feature the same numerical difficulties observed in reacting flow problems and allows to easily understand the very nature of them [4]. For this reason it has been considered as a first step in the developments of numerical methods for this class of problems also in [7,17,19].…”

“…The correct propagation speed of the reactive front may be restored by using a front-tracking approach, like the ghost fluid/level set method of Nguyen et al [6], by resorting to a local grid/time step refinement [7,8], that eliminates the local stiffness of the problem, or by a combination of both [9]. However, a very thin resolution is what one wants to avoid if only the general behavior of the fluid dynamical system and not the detailed investigation of the chemical phenomena is to be reproduced.…”

A split random time-stepping method for stiff and nonstiff detonation capturing

“…Geometry of the post-shock burnt gas part follows {|y − 0.5| > 0. 25 and the unburnt gas occupies the rest of domain before the initial shock. Initial states are identical with those in Example 6 except the x-velocity of the post-shock part is increased to u b = 2000m/s > u CJ , to create a strong detonation wave.…”

High Order Finite Difference Methods with Subcell Resolution for Stiff Multispecies Discontinuity Capturing

“…Therefore, many works have contributed to the analysis and development of underresolved numerical methods which are able to capture the correct shock/discontinuities location and speed without resolving the small chemical scales. Examples include the level set and front tracking methods [6,19,23,26,30], random choice method [10][11][12]24], random projection method [1][2][3] and many other works [4, 5, 7-9, 13, 14, 16, 17, 25, 27, 31, 32]. See Wang et al [33] for a comprehensive overview of the last two decades of this development.…”