A rarefaction-tracking method for hyperbolic conservation laws

Abstract

We present a numerical method for scalar conservation laws in one space dimension. The solution is approximated by local similarity solutions. While many commonly used approaches are based on shocks, the presented method uses rarefaction and compression waves. The solution is represented by particles that carry function values and move according to the method of characteristics. Between two neighboring particles, an interpolation is defined by an analytical similarity solution of the conservation law. An interaction of particles represents a collision of characteristics. The resulting shock is resolved by merging particles so that the total area under the function is conserved. The method is variation diminishing, nevertheless, it has no numerical dissipation away from shocks. Although shocks are not explicitly tracked, they can be located accurately. We present numerical examples, and outline specific applications and extensions of the approach. Comment: 21 pages, 7 figures. Similarity 2008 conference proceedings

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arXiv:0901.0298v2 [math.NA] 25 Jun 2009
A rarefaction-tracking method for hyperbolic
conservation laws
Yossi Farjoun Benjamin Seibold
Department of Mathematics
Massachusetts Institute of Technology
77 Massachusetts Avenue, Cambridge MA 02139
Abstract
We present a numerical method for scalar conservation laws in one
space dimension. The solution is approximated by local similarity so-
lutions. While many commonly used approaches are based on shocks,
the presented method uses rarefaction and compression waves. The so-
lution is represented by particles that carry function values and move
according to the method of characteristics. Between two neighboring par-
ticles, an interpolation is defined by an analytical similarity solution of
the conservation law. An interaction of particles represents a collision
of characteristics. The resulting shock is resolved by merging particles
so that the total area under the function is conserved. The method is
variation diminishing, nevertheless, it has no numerical dissipation away
from shocks. Although shocks are not explicitly tracked, they can be
located accurately. We present numerical examples, and outline specific
applications and extensions of the approach.
1 Introduction
Hyperbolic conservation laws are important models for the evolution of con-
tinuum quantities, and their efficient numerical approximation still involves
challenges. A fundamental property is the occurrence of shocks. These ap-
pear when parts of the solution that move at different characteristic velocities
collide, forming a traveling discontinuity. A sharp and accurate resolution of
shocks, without creating spurious oscillations, is one crucial challenge in nu-
merical methods. Traditional finite difference [17] or finite volume [8] methods
operate on a fixed grid. Straightforward linear numerical schemes are either
of low accuracy, or create overshoots near shocks [8]. These overshoots can be
avoided by considering more complex nonlinear schemes, such as finite volume
methods with limiters [24], and ENO [9] or WENO [20] schemes. These difficul-
ties show up in simple problems, such as scalar, one dimensional conservation
laws, even for linear advection. One interpretation of this challenge is that fixed
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grid methods attempt to represent advection by a change in function values.
However, since advection moves the solution along the x−axis, it is natural to
move the position of the computational nodes instead. Two popular approaches
that employ this either track shocks explicitly, or trace characteristics. One of
the first shock-tracking methods is Godunov’s scheme [8]. An initially piecewise
constant function is evolved analytically by solving the local Riemann problems
exactly. This works well until two shocks (or rarefactions) interact. Godunov’s
scheme avoids the resolution of this interaction by remeshing: the solution is
averaged over cells, and a new piecewise constant solution is reconstructed. In
this sense, Godunov’s scheme is actually a fixed-grid finite volume method, and
thus incurs the problems described above. The idea of tracing characteristics
has been formulated by Courant, Issacson and Rees in the CIR method [3]. The
solution of a conservation law is given by a simple ordinary differential equation
along a characteristic curve. The CIR method updates function values at grid
points by following characteristic curves. Function values between grid points
are defined by an interpolation, which is a remeshing step. Therefore, the CIR
method is a fixed grid approach as well. In the simplest case of using linear
interpolation between grid points, a simple non-conservative upwind scheme is
obtained.
Most popular numerical schemes for conservation laws are fixed grid ap-
proaches. Advantages include simple data structures and an easy generalization
to higher space dimensions using dimensional splitting. While these methods
can be derived from similarity solutions (finite volumes) or the method of char-
acteristics (CIR), they require a global remeshing step, which introduces numer-
ical errors, numerical dissipation and dispersion, and a decay in entropy even
away from shocks. More recent approaches attempt to avoid global remeshing,
or at least weaken its negative impact. An example of such an approach is
front tracking [13]. Shocks are tracked explicitly, but unlike with Godunov-type
schemes, no cells exist and no averaging is performed. Instead, the interaction
of the local similarity solutions is resolved where necessary. The difficulty of re-
solving interactions of rarefactions with shocks is avoided by approximating the
flux function by a piecewise linear function. Thus, only discontinuities have to
be tracked, and rarefactions are approximated by step functions. Analogously,
the method of characteristics can be used wherever characteristic curves do not
intersect. Lagrangian particle methods are based on this idea. Particles follow
the characteristics, and only when they come too close is their interaction re-
solved locally. Classical approaches, such as smoothed particle hydrodynamics
[22], consider a pressure that prevents particle collisions. While such approaches
work well in smooth regions, a stable treatment of shocks is challenging. In ad-
dition, hybrid methods exist, such as the self-adjusting grid method [10], which
operates on a fixed grid, and, in addition, tracks up to one shock particle in
each cell.
The method presented in the current paper can be interpreted as a combi-
nation of some of the aforementioned ideas. Characteristic curves are traced
using Lagrangian particles. In addition, between neighboring particles, a sim-
ilarity solution is evolved exactly. While front-tracking methods trace shocks,
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the current method uses rarefaction and compression waves to represent the
solution. Due to this analogy, the approach can be called rarefaction tracking.
The local similarity solutions give rise to a natural definition of area between
two particles. This, in turn, admits a merging strategy of colliding particles
that conserves area exactly. Thus, shocks are represented by compression waves
that are removed upon breaking. Due to the correct evolution of area, correct
shock speeds are obtained. The method was originally motivated in the context
of Lagrangian particle methods [6]. Well-posedness, accuracy and convergence
have been shown and investigated in [7]. This paper focuses on the connection
to local similarity solutions. In addition, applications for which the approach
has advantages over fixed-grid approaches are outlined. The method, as it is
presented here, is suited for scalar conservation laws, since for this kind of equa-
tions similarity solutions are straightforward to construct. Also, only one space
dimension is currently considered, since in this case a shock can only happen
if particles collide. However, extensions to balance laws with source terms are
presented, and other possible generalizations are outlined.
In Sect. 2, we outline the general characteristic particle approach. In Sect. 3,
similarity solutions between particles are constructed. These give rise to an ex-
act expression for area, which is the basis for particle management, as explained
in Sect. 4. Properties of the approach as presented in Sect. 5, and fundamen-
tal extensions (for simple sources, and flux functions with one inflection point)
are outlined in Sect. 6. In Sect. 7, results of computations with the presented
method are given. We consider an “academic” flux function, and two phys-
ical models. Possible applications of the method and possibilities for further
extensions are presented in Sect. 8.
2 The Particle Method
We consider a one dimensional scalar conservation law
ut+ f(u)x= 0,u(x,0) = u0(x) (1)
with f′continuous. In addition, until we explicitly relax the condition, we as-
sume that on the entire interval of considered function values [minxu(x),maxxu(x)],
f is either convex or concave. The characteristic equations [4]
?
˙ x = f′(u)
˙ u = 0
(2)
yield the solution (while it is smooth) forward in time: at each point (x0,u0(x0))
a characteristic curve x(t) = x0+ f′(u0(x0))t starts, carrying the function
value u(x(t),t) = u0(x0). The method of characteristics gives rise to a particle
method:
1. Sample the initial function u0(x) by a finite number of points (xi,ui). The
aspect of sampling the initial conditions is addressed in Sect. 3.2.
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2. The solution at time t > 0 is given by the points (xi+f′(ui)t,ui). At these
points, the solution is represented exactly, without any approximation
error.
This basic approach incurs two fundamental limitations, which need to be reme-
died before it can become a useful method:
• The solution is only known at the particle positions. If neighboring par-
ticles depart, large regions of unspecified function value may arise. The
presented approach remedies this problem by defining an interpolation be-
tween particles. This allows an easy insertion of new particles into gaps
wherever desired.
• A particle may overtake its neighbor, corresponding to the intersection of
characteristic curves. The correct weak solution of the conservation law (1)
develops a shock, i.e. a moving discontinuity, into which the characteristic
curves keep running [4]. In the presented approach, particles are merged
upon collision. An interpolation between particles allows the merging of
particles in such a way that area is conserved.
The insertion and merging of particles is called particle management.
described in detail in Sect. 4. Particle management is the key ingredient that
enables us to use the method of characteristics beyond the first occurrence of
shocks.
Given the solution at time t, represented by particles (xi(t),ui(t)), the time
of the next particle collision is t + ∆ts, where
It is
∆ti= −
xi+1− xi
f′(ui+1) − f′(ui)
(3)
is the time until the collision between particles i and i + 1, and
∆ts= min{{∆ti|∆ti≥ 0} ∪ ∞}(4)
is the time until the next collision in the future. We can step forward by up
to ∆ts without risking a collision. After ∆ts, at least one particle will share
its position with another. To proceed further, we need to perform particle
management. We insert new particles into gaps between particles of distance
larger than a maximum distance dmax. Then, each pair of particles sharing their
position is merged. Optionally, one can perform a merging step for all particles
closer than a minimum distance dmin. Choices for the maximum and minimum
distance are described in Sect. 4.
3Interpolation by Similarity Waves
Consider two neighboring particles (x1(t),u1), (x2(t),u2), with x1 < x2. If
f′(u1) > f′(u2), the two particles meet in the future, at time t + ∆t1, where
∆t1 > 0 is given by (3). Let us assume that between the two particles, the
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solution is continuous at all times before t+∆t1. This implies that the meeting
of the particles is the first occurrence of a shock, and at this time the solu-
tion is a discontinuity at position xsh = x1+ f′(u1)∆t1. Any point (xsh,u),
u ∈ [u1,u2] on the shock can be traced backwards along its characteristic to
(xsh− f′(u)∆t1,u). As shown in Fig. 1, this defines the interpolation uniquely
as
x(u) = x1+ f′(u1)∆t1− f′(u)∆t1= x1+f′(u) − f′(u1)
f′(u2) − f′(u1)(x2− x1) . (5)
In the case of departing particles, i.e. f′(u1) < f′(u2), the above construction
is exactly reversed in space and time. We assume that the solution between
the particles came from a discontinuity at some time in the past. Consequently,
formula (5) also describes this case.
Theorem 1. The interpolation (5) between two particles is a solution of the
conservation law (1). Consequently, for multiple particles, the function defined
piecewise by the interpolation (5) is a classical (i.e. continuous) solution to the
conservation law (1), until particles collide.
Proof. Using that ˙ xi(t) = f′(ui) for i = 1,2 (from (2)) one obtains
∂x
∂t(u,t) = ˙ x1+f′(u) − f′(u1)
= f′(u1) +f′(u) − f′(u1)
f′(u2) − f′(u1)(˙ x2− ˙ x1)
f′(u2) − f′(u1)(f′(u2) − f′(u1)) = f′(u) .
Thus every point on the interpolation u(x,t) satisfies the characteristic equation
(2). At the particles, the solution possesses kinks. These can be interpreted as
shocks of height zero, and thus satisfy the Rankine-Hugoniot conditions [4].
Hence, the full piecewise wave solution is a weak solution of (1).
The interpolation (5) defines either a rarefaction wave (if f′(u1) < f′(u2)),
or a compression wave (if f′(u1) > f′(u2)) between two particles. In the case
u1= u2, a constant interpolant is defined. Hence, at all times the solution is
approximated by similarity waves.
Remark 1. That (5) defines the interpolant as a function x(u) is, in fact, an
advantage, since at a discontinuity x1= x2, characteristics for all intermediate
values u are defined. Thus, rarefaction fans arise naturally from discontinuities
if f′(u1) < f′(u2). If f has no inflection points between u1and u2, the inverse
function u(x) exists. For plotting purposes we plot x(u) instead of inverting the
function.
3.1Evolution of Area
The interpolation defines an area between particles. As shown in [7], the inte-
gration of (5) yields
?x2(t)
x1(t)
u(x,t)dx = (x2(t) − x1(t))a(u1,u2) , (6)
5