The main contributions of this chapter are novel numerical reachability analysis methods for hyperbolic equa- tions, a particular class of PDEs. We develop numerical reachability analysis for linear/nonlinear hyperbolic equations by combining the flux splitting method, theLax-Friedrichmethod, and theLax-Wendroff method with optimization and interpolation. These methods provide ways of approximating mesh values and estimat- ing reachable sets on the points. In linear systems, the estimated reachable sets are constructed by computing the trajectories using extreme values from the initial sets. While, in nonlinear systems, an optimization prob- lem is proposed at each time step to calculate the bounds of the values for the next time step. Eventually, the sets are built through interpolation with respect to both time and space, when the values at the current and subsequent time steps are known. In the example section, the estimated reachable sets are compared with real reachable sets. Through dense spatial discretization, the estimated reachable sets can provide a good approximation of the real ones.
4.2.1 Problem Formulation
The following class of systems is considered:
ut+f(u)x=0,t≥0, x∈D⊆R u(x,0) =ϕ(x)
, (4.1)
whereu= (u1,u2,u3, ...,uk)T,ut= (∂u1
∂t ,∂u1
∂t , ...,∂uk
∂t )T, f(u) = (f1(u),f2(u), ...,fk(u))T is the flux function, f(u)x= (∂f1(u)
∂x ,∂f2(u)
∂x , ...,∂fk(u)
∂x )T,ϕ(x) = (ϕ1(x),ϕ2(x), ...,ϕk(x))T is the initial condition, D⊆Ris the domain of interest andu:D×(R+∪0)→Rk.
The initial condition is represented byϕ(x),ϕ:D7→Rk, and is constrained by a range[ϕl(x),ϕu(x)], whereϕl(x)is the lower bound ofϕ(x)forx∈D andϕu(x)is the upper bound ofϕ(x)for x∈D. The lower and upper bounds compose the initial set of the problem described in (4.1). The definition, used in this chapter, for initial sets is:
Definition 22(Initial Set). Letϕ(x)be the initial condition of (4.1) and ϕl(x)and ϕu(x) are the corre- sponding lower bound and upper bound ofϕ(x)defined on domainD. The initial set of (4.1) is defined as
Ω≜ (
[
x
[ϕl(x),ϕu(x)]| for x∈D )
. (4.2)
Reachable sets are shown to start fromΩand evolve accordingly.
Definition 23(Reachable Set). The reachable set of (4.1) is
Ru(T)≜ (
[
x,t
u(x,t)|u(x,t)satisfy(4.1)for x∈D,t∈[0,T] )
. (4.3)
Since PDEs are often nonlinear and unsolvable [177], it is almost impossible to compute the exact reach- able setRu(T)for a system of PDEs. Rather than directly computing the reachable set, a more practical and feasible way is to derive an approximation ofRu(T), which is called reachable set estimation. In this chapter, the estimation is based on the finite difference methods.
Finite difference methods (FDMs) are a group of approaches used for approximating real values or vari- able derivatives of a function on predefined mesh points. The error between the discrete solution and the exact solution is measured through a Taylor expansion. We show how to construct estimated reachable sets R˜u(T)after discussing FDMs and pick the most accurate estimation forRu(T).
The main focus of this work is on safety verification for hyperbolic systems. The safety specification is expressed by a setS, defined on domainD, and describes the safety requirement.
Definition 24. Consider a hyperbolic PDE system in the form of (4.1), reachable sets Ru(T)and a safety specification S, the hyperbolic PDE system is safe if the following condition is satisfied
Ru(T)∩ ¬S=/0. (4.4)
Definition 24 uses reachable sets for safety verification of a system of PDEs althoughRu(T)is not obtain-
able due to the unsolvable property mentioned above. One way to circumvent these difficulties is to utilize FDMs since it produces an explicit error estimation. As the number of mesh points increases, the estimated solution will become more accurate. In the next section, the analysis of linear and nonlinear systems is discussed.
4.2.2 Linear System
The flux function f(u)in (4.1) can be decomposed if its Jacobian matrix is not dependent on the unknown variableu. Then, we obtain the system below.
ut+Aux=0,t∈R+, x∈D⊆R u(x,0) =ϕ(x)
(4.5)
whereAis the Jacobian matrix of f(u)and is a constant matrix.
Transforming the matrixAinto a diagonal matrixΛallows eachu∈uto be treated separately. Next,Sis multiplied on both sides ofut+Aux=0.
Sut+SAux=0, (Su)t+SAS−1Sux=0,
(Su)t+Λ(Su)x=0.
(4.6)
Replacing(Su)tbyvt and(Su)xbyvx, we have
vt+Λvx=0. (4.7)
In order to perform reachability analysis, we define monotone to simplify the computing. If our iteration procedure is monotone, we only need to simulate 2 trajectories to construct the reachable sets. An additional CFLcondition is required for the monotone property.
4.2.3 Nonlinear System
In the general form shown in system 4.1, if the Jacobian matrixAis a function ofuandf(u)x=A(u)ux, then it is a nonlinear system.
A commonly used approach for simulating a system is theLax-Wendroff method [182] pages 213–216.
It is second-order accurate with respect to both space and time. The main issue with numerical reachability analysis using theLax-Wendroff method is that it is not monotone and does not preserve the property. That
fact has been proved by Harten [84]. In order to handle the non-monotone issue, an optimization problem is constructed and solved at each time step. The maximum and minimum values at each mesh point are found and used as the starting values for the next time step.
The goal is to findun+1min,i andun+1max,i. To find the optimal solution at timetn, start with the mesh point iat timetnand set up the optimizations described below generally using the optimization operator opt∈ {min,max}:
optun+1i
s.t.Lax-Wendroff scheme, uni−1∈[unmin,i−1, unmax,i−1], uni ∈[unmin,i, unmax,i], uni+1∈[unmin,i+1, unmax,i+1].
(4.8)
4.3 Numerical Reachability Analysis for Parabolic Equations with Dense Spatial Discretization