The random projection method for a model problem of combustion with stiff chemical reactions

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Abstract

In this paper we extend the random projection method, recently proposed by the author and S. Jin [J. Comput. Phys. 163 (2000) 216] for under resolved numerical simulations of a qualitative model problem for combustion with stiff chemical reactions:ut+(f(u)−q0z)x=0,zx=1εφ(u)z.In this problem, the reaction time ε is small, making the problem numerically stiff. A classic spurious numerical phenomenon – the incorrect shock speed – occurs when the reaction time scale is not properly resolved numerically. The random projection method is introduced recently to handle this kind of numerical difficulty. The key idea in this method is to randomize the ignition temperature in a suitable domain. Several numerical experiments demonstrate the reliability and robustness of this method.

Introduction

Consider a model problem for dynamic combustion with chemical reactions:ut+(f(u)−q0z)x=0,zx=1εφ(u)z,where u is a lumped variable representing some functions of density, velocity, temperature and f is a convex function of u, z can be thought of as describing the mass fraction of unburnt gas, 1/ε is the rate of reaction, q0 represents the amount of unit energy liberated by chemical reaction (in particular, q0>0 for exothermic reactions) and φ(u) is typically a smooth increasing function with 0⩽φ⩽1 describing how the reaction rate depends on u. In this paper, we considerφ(u)=1,u>um,0,u⩽um,where um can be thought of as an ignition temperature.

This is a qualitative model, introduced by Majda [15], for dynamic combustion with chemical reactions. In this model, the reacting time ε is much smaller than the characteristic time scale of the fluid, making the reaction term in (1.2) stiff.

Numerical methods for this kind of problems have attracted a great deal of attention in the last decade. It was first observed by Colella, Majda and Roytburd [8] that an under resolved numerical method, where ε is not resolved by suitable small time steps and grid sizes, leads to spurious shock that travels one grid per time step. Since then, lots of attention have been paid to study this peculiar numerical phenomenon (see [3], [4], [14], [16] and references therein) and or find a robust way to fix it (see [9], [10]). It is known that numerical shock profile, an essential mechanism in all shock capturing method, leads to too early chemical reaction once the smeared value in the numerical shock layer is above the ignition temperature (um in (1.3)).

Recently the author and Jin proposed the random projection method as a general and systematic method to solve hyperbolic systems with stiff source terms, applicable to reacting flow problems [1]. Unlike the classical random choice method for reacting flow [6], originated from Glimm's scheme [5], [11], which requires solving a generalized Riemann problem for hyperbolic systems with source terms. The random projection method we proposed is a fractional step method that combines a standard – no Riemann solver or generalized Riemann solver is needed – shock capturing method for the homogeneous convection with a strikingly simple random projection step for the reaction term. In the random projection step, the ignition temperature is chosen to be a uniformly distributed random variable between two stable equilibria. Although at each time step, this random projection will move the shock with incorrect speed, the statistical average yields the correct speed, even though the small time scale ε is not numerically resolved. In particular, when the random number is chosen to be the equidistributed van der Corput sampling sequence [12], we proved, for a model scalar problem, a first order accuracy on the shock speed if a monotonicity-preserving method – which includes all TVD schemes – is used in the convection step [2]. A large amount of numerical experiments demonstrate the robustness of this novel approach.

In this paper, we extend the random projection method for the qualitative model for combustion with chemical reactions , . Several numerical experiments will be conducted to examine the effectiveness and robustness of the method.

The paper is organized as follows: In Section 2 we introduce the random projection method for the problem , . In Section 3 several numerical examples will be presented. We end in Section 4 with some concluding remarks.

Section snippets

The random projection method

In this section, we shall describe the random projection method for , with piecewise constant initial datau(x,0)=u0(x)=ul,x⩽x0,ur,x>x0,z(x,0)=z0(x)=1,x⩽x0,0,x>x0and boundary conditionlimx→+∞z(x,t)=1.Without loss of generality the data in , , are chosen such that the discontinuity, initially at x=x0, moves to the right. The case when the discontinuity moves to the left can be treated similarly.

Let the grid points be xi, i=…,−1,0,1,…, with equal mesh spacing h=xi+1xi. The time level t0=0,t1,t2

Numerical examples

In order to verify the performance of the random projection method for the qualitative model of combustion with stiff chemical reactions, we conduct several numerical experiments. Among the numerical examples, the operator Sc(k) is chosen as the second order relaxed scheme [13], which is a TVD scheme without the usage of Riemann solvers or local characteristic decompositions.

Example 1

We choose f(u)=12u2 and q0=2.2 in (1.1), ε=10−4 in (1.2), um=−1.45 in (1.3), ul=1.363325, ur=−1.5 in (2.1) and x0=0.1 in

Conclusions

In this paper, we extend the random projection method to under resolved computation of a qualitative model for combustion with stiff chemical reactions. This method is based on the random projection method proposed by the author and Jin for general hyperbolic systems with stiff source terms [1]. The key idea of this method is to randomize the ignition temperature in the two equilibrium states. Numerical experiments demonstrate that this method, although very simple and efficient, provides

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