THE ALTERNATING EVOLUTION METHODS FOR FIRST ORDER NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

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1 THE ALTERNATING EVOLUTION METHODS FOR FIRST ORDER NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS HAILIANG LIU TO MARSHALL SLEMROD ON HIS 7TH BIRTHDAY WITH FRIENDSHIP AND APPRECIATION. Abstract. In this paper, we present a brief survey of the recent developments in alternating evolution (AE) methods for numerical computation of first order partial differential equations, with hyperbolic conservation laws and Hamilton-Jacobi equations as two canonical examples. The main difficulty of such computation arises from the nonlinearity of the model, making it necessary to incorporate an appropriate amount of numerical viscosity to capture the entropy/viscosity solution as physically relevant solutions. The alternating evolution method is based on the AE system of the original PDEs, the discretization technique ranges from finite difference, finite volume and the discontinuous Galerkin methods. In all these cases, the AE solver can produce accurate solutions with equal computational time than the traditional solvers. In particular, the AE formulation allows the same discontinuous Galerkin discretization for both conservative and non-conservative PDEs under consideration. In order to make the presentation more concise and to highlight the main ideas of the algorithm, we use simplified models to describe the details of the AE method. Sample simulation results on a few models are also given.. Introduction In mathematics, a first-order partial differential equation (PDE) is an equation that involves only first order derivatives of the unknown function of multiple variables. The equation may take the form F (ξ,φ, ξ φ)=. Such equations arise in the construction of characteristic surfaces for hyperbolic partial differential equations, in the calculus of variations, in some geometrical problems, and in simple models for gas dynamics whose solution involves the method of characteristics. For the resolutions of the underlying application problems, it has been proven to be important to analyze and solve the governing PDEs. For time dependent problems we distinguish the temporal variable t> from the spatial variable x R d, and a first order PDE may be symbolically written as t φ + A(φ) =, x R d, t >, where A is a nonlinear differential operator. Two classes of PDEs are of particular interest: hyperbolic conservation laws A = x f(φ) and Hamilton-Jacobi equations A = H(x, x φ), which will be used to illustrate the main ideas of the alternation evolution (AE) methods in this article. Date: September,3. 99 Mathematics Subject Classification. 35K5, 65M5, 65M6, 76R5. Key words and phrases. Conservation laws, Hamilton-Jacobi equations, Alternating evolution methods, Lax- Friedrichs.

2 HAILIANG LIU TO MARSHALL SLEMROD ON HIS 7TH BIRTHDAY WITH FRIENDSHIP AND APPRECIATION... Hyperbolic conservation laws A = x f(φ). A multi-dimensional hyperbolic conservation law has the form: () t φ + x f(φ) =, x R d, t >, where φ R m denotes a vector of conserved quantities, and f : R m R d is a nonlinear convection flux. The compressible Euler equation in gas dynamics is a canonical example. These equations are of great practical importance since they model a variety of physical phenomena that appear in fluid mechanics, astrophysics, groundwater flow, traffic flow, semiconductor device simulation, and magneto-hydrodynamics, among many others. The notorious difficulty encountered for the satisfactory approximation of the exact solutions of these systems lies in the presence of discontinuities in the solution, leading to non-uniqueness of the weak solution. We are interested in computing the physically relevant solution so called the entropy solution. The entropy solution φ is defined so that φ satisfies the entropy inequalities () (η(φ) t v + q(φ) x v)dxdt R d for each v C (Rd (, ) and v and each entropy/entropy flux pair (η, q): η is convex and Dη(z) Df j (z) =Dq j (z), j = d... Hamilton-Jacobi equations A = H(x, x φ). Another important class is the Hamilton- Jacobi equation: (3) t φ + H(x, x φ)=, φ(x, ) = φ (x), x R d, t >. Here the unknown φ is scalar, and H : R d R is a nonlinear Hamiltonian. The Hamilton- Jacobi equation arises in many applications ranging from geometrical optics to differential games. These nonlinear equations typically develop discontinuous derivatives even with smooth initial conditions, such weak solutions are not unique. It would be natural to try to extend the entropy solution concept based on integration by parts in () to Hamilton-Jacobi equations, unfortunately, attempts as such had gone fruitless until the first breakthrough in 983 when Crandall and Lions introduced the notion of viscosity solutions and established their theory for Hamilton-Jacobi equations, see [9, 5, 59]. In the concept of viscosity solutions, the derivative of the solution is compared to a fixed test function at certain points via the comparison principle. A bounded uniformly continuous function φ is called a viscosity subsolution (supersolution) of (3) if, for every point (x,t ) and a function v C satisfying φ ( )v and φ(x,t )=v(x,t ), there holds t v + H(x, x v) at (x,t ). Moreover, φ is called a viscosity solution if φ is simultaneously a viscosity subsolution and a viscosity supersolution. We are interested in computation of the viscosity solution, which is the unique physically relevant solution in some important applications. The difficulty encountered for the satisfactory approximation of the exact solutions of these equations lies in the presence of discontinuities in the solution derivatives..3. Numerical methods. Numerical discretization methods for first order PDEs are diverse, the popular ones are finite difference methods, finite volume methods and the discontinuous Galerkin methods. Among the considerable amount of literature available on numerical methods for hyperbolic conservation laws and Hamilton-Jacobi equations, the discretization solution techniques fall under two main categories according to their way of sampling [5]: upwind and central schemes.

3 THE AE METHOD 3 The forerunners for these two classes of high resolution schemes for conservation laws are the first order Godunov [7] and Lax-Friedrichs (LxF) schemes [6, 33], respectively. The need for devising more accurate and efficient numerical methods for conservation laws and related models has prompted and sustained the abundant research in this area, see, for example, [34, 6, 6, 7]. The success of high resolution schemes has been due to two factors: the local enforcement of the underlying PDE and the non-oscillatory piecewise polynomial reconstruction from evolved local moments (cell averages or gird values). For conservation laws, various higher-order extensions of the Godunov type finite volume scheme have been rapidly developed since 97 s, employing higher-order reconstruction of piecewise polynomials from the cell averages, including MUSCL, TVD, PPM, ENO and WENO schemes [6, 63, 9, 8,, 57, 58, 44]. In this development, the local refinement of one dimensional conservation laws may be expressed as (4) t φ + x ˆf(φ,φ + )=, where ˆf is an entropy satisfying numerical flux. For Hamilton-Jacobi equations, the ENO/WENO type schemes ([4, 35, 54, 55, 56, 53, 65]) are based mainly on some local refinement of H-J equations by (5) t φ + Ĥ(x, φ+ x,φ x )=, where Ĥ is the numerical Hamiltonian which needs to be carefully chosen to ensure that the viscosity solution is captured when φ x becomes discontinuous. In contrast, central type schemes such as the LxF scheme are more diffusive, yet easy to formulate and implement since no Riemann solvers are required. Examples of such schemes for conservation laws are the second-order Nessyahu-Tadmor scheme [5] and other higher-order schemes in [45,, 36, 5, 3, 3]. For H-J equations, the central type schemes ( [, 3, 4, 9, 3, 38, 39]) choose to evolve the constructed polynomials in smooth regions so that the Taylor expansion may be used in the scheme derivation. The two categories of the existing high resolution schemes are somehow interlaced during their independent developments; the upwind scheme becomes Riemann solver-free when a local numerical flux can be identified to replace the exact Riemann solver, see Shu and Osher [57, 58], and the central scheme becomes less diffusive when variable control volumes are used in deriving the scheme, see Kurganov and Tadmor [3]. The upwind feature can be further enforced [, 9] in central-upwind schemes, see [8] for a recent derivation of such a scheme. The relaxation scheme of Jin and Xin [6] provides yet another approach for solving nonlinear conservation laws, see also [7, 5]. The discontinuous Galerkin (DG) method has the advantage of flexibility for arbitrarily unstructured meshes, and with the ability to easily achieve arbitrary order of accuracy. The DG method has been quite successful for conservation laws [,,, 3, 4] due to the conservative nature of the formulation (4). However, new difficulties occur when the existing ideas with finite difference methods are applied toward the discontinuous Galerkin discretization for the Hamilton-Jacobi equation (5). One main difficulty comes from the non-conservative form of (5), which precludes the use of integration by parts to establish the cell to cell communication via numerical fluxes as usually done with the DG methods for conservation laws. In spite of this difficulty, some progress has been made in past years, see [, 6, 37, 64]. With the alternating evolution framework reviewed in this article, one can apply the same DG discretization to both conservation laws and Hamilton-Jacobi equations..4. The alternating evolution (AE) system. The difference between the AE schemes discussed in this article and the existing schemes mentioned above lies in the local enforcement of

4 4 HAILIANG LIU TO MARSHALL SLEMROD ON HIS 7TH BIRTHDAY WITH FRIENDSHIP AND APPRECIATION. underlying PDEs. Instead of using either (4) or (5), we refine the original PDE by an alternating evolution (AE) system t u + Ã(u, v) = (v u), t v + Ã(v, u) = (u v), which involves two representatives: {u, v}. Here à is a refinement of A so that terms involving spatial derivatives are replaced by v s derivatives, the additional relaxation term (v u)/, serves to communicate the two representatives u and v, with> being a scale parameter of user s choice. This amount of leverage in the choice of adds another attractive feature of the AE scheme. Indeed different choices of the scale parameter in such a procedure yield different AE schemes [4]. We have Ã(u, v) = x f(v) for conservation laws, and Ã(u, v) =H(x, xv) for Hamilton-Jacobi equations. The AE system for scalar hyperbolic conservation laws was originally proposed in [4], where the system was shown to be capable of capturing the exact solution when initially both representatives are chosen as the given initial data. Such a feature allows for a sampling of two representatives over alternating grids/cells. Using this alternating system as a building base, we apply standard approximation techniques to the AE system: high order accuracy is achieved by a combination of high-order non-oscillatory polynomial approximation in space and an ODE solver in time with matching accuracy. For conservation laws we sample using cell averages [4, 4] and for Hamilton-Jacobi equation we sample using grid values [4], which correspond to finite volume and finite difference schemes, respectively. For the discontinuous Galerkin discretization, same sampling of local moments is applied to both types of equations [43]. The procedure using the AE formulation opens a new way to derive robust highly accurate schemes for nonlinear PDEs, conservative or non-conservative. The AE schemes are very easy to implement and efficient. Actually such simplicity in implementation is even more pronounced in the multi-dimensional case. More closely related to the AE scheme is the overlapping cell schemes introduced by Liu [46], who generalizes the NT scheme [5] by evolving two pieces of information over redundant overlapping cells, therefore allows easy formulation of semi-discrete schemes. The technique has been extended to the development of some central discontinuous Galerkin methods for conservation laws and diffusion equations [47, 48, 49, 5], as well as for Hamilton-Jacobi equations [37]. One main difference between the overlapping central schemes and the AE schemes is that central overlapping schemes use two polynomial representatives solved on two sets of overlapping meshes, and the AE schemes only have one polynomial representative associated with each grid point, even in multi-dimensional case. The advantage of the AE scheme is clearer in the multi-dimensional case. It should be noted that even though our AE schemes are derived based on sampling the alternating evolution system, we do not solve the system directly. The AE system simply provides a systematic way for developing numerical schemes of both semi-discrete and fully discrete form for the underlying PDE, instead of as an approximation system at the continuous level. The remainder of the article is organized as follows. In section, we illustrate how to derive the alternating evolution system from the Lax-Friedrichs scheme for one-dimensional conservation laws. We formulate the high resolution finite volume AE schemes for hyperbolic conservation laws ([4]) in section 3, and high resolution finite difference AE schemes for Hamilton-Jacobi equations ([4]) in section 4. In section 5, we present high order AEDG schemes for both Hamilton-Jacobi equations ([43]) and hyperbolic conservation laws. Sample simulation results

5 THE AE METHOD 5 on a few models are given in section 6. The last section 7 ends this paper with our concluding remarks.. From the Lax-Friedrichs scheme to the AE system The AE system was motivated by the Lax-Friedrichs scheme, as described in [4]. We start with finite difference schemes for one dimensional scalar hyperbolic conservation laws (6) t U + x f(u) =. Let the xt-plane be covered by a rectangular gird with mesh size x in x direction and t in the t direction. It would seem natural to replace (6) by the difference equation u n+ j = u n j λ f(u n j+ ) f(u n j ), where λ = t/ x and u n j approximates U(j x, n t). But this is known to be inappropriate because of the high degree of instability of this scheme. Replacing u n j by the average of its two neighbors u n j± we encounter the celebrated Lax-Friedrichs (LxF) scheme (7) u n+ j = un j+ + un j λ f(u n j+ ) f(u n j ). One noticeable feature of this scheme is that information on grids j + n = even is independent of that on grids j + n = odd. Rewriting (7) as u n+ j u n j (8) + f(u n t x j+ ) f(u n j ) u n j+ +un j u n j =. t If we denote the solution at even (or odd) grid points as u and those at odd (or even) girds as v, and bring in a scale parameter t, then u n+ j u n j (9) + f(v n t x j+ ) f(vj ) n vj+ n +vn j u n j =. Passing to the limit x, t in (9) and keeping the parameter unchanged, we obtain a coupled system for both u and v, which in multi-dimensional case can be expressed as () () t u + x f(v) = (v u), x t v + x f(u) = (u v). Rd The main feature of this model for the system of conservation laws (6) is its high accuracy. The convergence for scalar conservation laws with general smooth flux function in arbitrary spatial dimension R d is given in ([4]), and summarized in the following. Theorem.. For any (u,v ) L (R d ) L (R d ) and each fixed, let ()-() admit a unique weak solution (u,v ) on R d R + such that (u, v) C([, ); L (R d )), then there exists a bounded measurable function U(x, t) on R d R + such that as u U(x, t), u v (x, t) R d R +. Moreover, U is the entropy solution of (6) with initial data U (x) = (u (x)+v (x)) for x R d. Corollary.. Let U be the entropy solution of the scalar conservation laws (6) with initial data U L (R d ) L (R d ),and(u,v ) be the weak solution to ()-() subject to initial data with (u,v ) L (R d ). Then it holds u v L (R d ) u v L (R d )e 4t/.

6 6 HAILIANG LIU TO MARSHALL SLEMROD ON HIS 7TH BIRTHDAY WITH FRIENDSHIP AND APPRECIATION. Furthermore, (i)if u + v =U, then (ii) if u = v = U, then lim t/ u (,t) U(,t) L (R d ) = lim t/ v (,t) U(,t) L (R d ) =. almost everywhere in R d R +. u (x, t) =v (x, t) =U(x, t) In an entirely similar manner, one may obtain a coupled alternating evolution system () (3) t u + H(x, u, x v)= (v u), t v + H(x, v, x u)= (u v), as a refinement of the Hamilton-Jacobi equation (4) t φ + H(x, φ, x φ)=. Based on the AE system both high resolution finite difference method and the discontinuous Galerkin method are designed for solving the Hamilton-Jacobi equation. 3. Finite volume schemes Hyperbolic conservation laws Consider the system of hyperbolic conservation laws (5) t u + x f(u) =, (x, t) R d (, ), where u = (u,...,u m ) T. In multi-dimensional case, for simplicity, we take uniform distributed grids at x α with multi-index α =(α,...,α d ). Let I α be a rectangle with vertices at {x α+β, β =}, labeled as x α±, the number of which amounts to d. We take the average of the AE equation, over I α to obtain where ū α = I α I α u(x, t)dx and t u + x f(v) = (v u), d L α [v] = I α dtūα + ūα = L α[v](t), I α vdx f(v) ν α ds. I α I α Here, I α denotes the volume of I α, I α indicates the boundary of I α, and ν α is the outward pointing unit normal field of the cell boundary I α. Let Φ α ū α denote the numerical solution, and reconstruct non-oscillatory polynomials p α [Φ] over I α from available averages Φ α,thenwe obtain a semi-discrete scheme d (6) dt Φ α + Φ α = L α[φ], where p SN α [Φ] s constructed over neighboring cells I α± are to be used to evaluate (7) L α [Φ] = p SN α [Φ](x) dx f(p SN α [Φ](x)) ds. I α I α I α± I α I α I α±

7 THE AE METHOD 7 Finally, in order to obtain the same order accuracy in time, the semi-discrete (6)-(7) is to be complemented with an ODE solver with matching accuracy in time discretization. The evolution parameter is chosen such that (8) d j= max f j ( ) x j Q, t <, where Q depends on the order of the scheme, see (3) or (3) later in this section. For system case, fj s need to be replaced by the dominant eigenvalues over a Riemann curve. For the D setting, we illustrate the corresponding AE scheme for (9) t φ + f(φ) x + g(φ) y =. 3.. First order scheme. If the reconstructed polynomial is piecewise constant, then we obtain the first order scheme as () Φ n+ k,l = ( κ)φ n k,l + κl k,l[φ n ], where upon a direct calculation using p k,l [Φ n ]= Φ n χ k,l I kl (x, y), where χ I kl is the characteristic function which takes value one on the rectangle I k,l, L k,l [Φ n ] can be expressed as L k,l [Φ n ] = A x A y Φ n k,l f(φ n 4 x k+,l ) f(φ n k,l )+f(φn k+,l+ ) f(φn k,l+ ) g(φ n () 4 y k,l+ ) g(φ n k,l )+g(φn k+,l+ ) g(φn k+,l ), with κ = t,wherea x and A y are the average operators in both x and y direction, respectively, so that () A x A y Φ n k,l := 4 Φ n k,l +Φ n k+,l +Φn k+,l+ +Φn k,l+. The parameter is chosen so that max f (3) x which ensures the scalar maximum principle (4) + max g and t <, y Φ n+ Φ n, n N. Again, for system case, both f and g in the stability requirement (3) needs to be replaced by dominant eigenvalues of the corresponding Jacobian matrices. 3.. Second order scheme. The second order scheme requires a linear polynomial reconstruction, its formulation in the rectangle I kl =[x k x, x k + x] [y k y, y k + y] has the form (5) p k,l [Φ n ](x, y) = k,l Φ n k,l + s k,l (x x k)+s k,l (y y l) χ Ikl (x, y), where s and s are the numerical derivatives corresponding to Φ x and Φ y. Now, using the midpoint quadrature rule in evaluation of (7), we obtain the second order AE scheme d dt Φ k,l = L k,l [Φ]

8 8 HAILIANG LIU TO MARSHALL SLEMROD ON HIS 7TH BIRTHDAY WITH FRIENDSHIP AND APPRECIATION. with L k,l [Φ] = A x A y Φ k,l + x s 8 k,l s k+,l s k+,l+ + k,l+ s + y s 8 k,l + s k+,l s k+,l+ k,l+ s f Φ k+,l + y 4 x s k+,l ) f Φ k,l + y s k,l ) +f Φ k+,l+ y s k+,l+ ) f Φ k,l+ y s k,l+ ) g Φ k,l+ + x 4 y s k,l+ ) g Φ k,l + x s k,l ) +g Φ k+,l+ x s k+,l+ ) g Φ k+,l x s k+,l ). The non-oscillatory property requires that we choose s k,l and s k,l with certain limiters. In [4] the basic minmod limiter is adopted: Φ n s k+,l Φ n k,l k,l = minmod, Φn k,l Φn k,l (6), x x Φ n s k,l+ Φ n k,l k,l = minmod, Φn k,l Φn k,l (7), y y with (8) minmod {a,a,...} = min{a i } if a i > i, i max{a i } if a i < i, i otherwise. When the second order Runge-Kutta time discretization is used, the scheme becomes (9) (3) Φ k,l = ( κ)φ n k,l + κl k,l[φ n ], Φ n+ k,l = Φn k,l + ( κ)φ k,l + κ L k,l[φ ]. Indeed analysis in [4] shows that such a choice again yields the scalar maximum principle (4), provided max f + max g (3) and t <. x y 4 Even higher order schemes can be constructed using the ENO selection of further stencils, see [4] for further details. 4. Finite difference schemes Hamilton-Jacobi equations To approximate the multi-dimensional HJ equations: we start with the AE formulation t φ + H(x, x φ)=, x R d, (3) t u + u = v H(x, xv). Again we use {x α } to denote uniformly distributed grids in R d, and I α as a hypercube centered at x α with vertices at x α± where the number of vertices is d.

9 THE AE METHOD 9 In [4] we present two different constructions of AE schemes for multi-dimensions. For the first type, given grid values {Φ α }, we construct a continuous, piecewise polynomial p α [Φ](x) P r defined in I α such that p α [Φ](x α± )=Φ α±. Here P r denotes a linear space of all polynomials of degree at most r in all x i : P r := {p p(x) = a β (x x α ) β, i d, a β R}. β i r Sampling the AE system (3) at x α, which is the common vertex of I α±,whileusingthese polynomials on the right hand side of (3), we obtain the semi-discrete AE scheme (33) d dt Φ α + Φ α = L α[φ], L α [Φ] = p SN α [Φ](x α ) H(x α, x p SN α [Φ](x α )), where p SN α [Φ] s are constructed using neighboring grid values. Using this sampling approach, construction of schemes of higher than second order will become cumbersome. In [4] we presented a better approach of reconstruction, called the dimension-by-dimension approach. Such an approach can be easily derived for higher order schemes, and work equally well with hyperbolic conservation laws. We illustrate the approach in two dimensional setting. We construct interpolated polynomials, p j,k and q j,k in the x and y direction as in onedimensional case, satisfying p j,k [Φ](x j±,y k )=Φ j±,k, q j,k [Φ](x j,y k± )=Φ j,y±, so that for Hamiltonian H = H( x φ)wehave (34) L j,k = p j,k[φ](x j,y k )+q j,k [Φ](x j,y k ) H ( x p j,k [Φ](x j,y k ), y q j,k [Φ](x j,y k )). 4.. First order scheme. For the first order scheme, such an interpolant is given by where p j,k [Φ](x, y) = Φ j,k + s x j,k (x x j ), q j,k [Φ](x, y) = Φ j,k + s y j,k (y y k ), s x j,k = Φ j+,k Φ j,k x, s y j,k = Φ j,k+ Φ j,k. y Evaluating at the polynomials and their partial derivatives at (x j,y k )yields p j,k [Φ](x j,y k ) = Φ j+,k +Φ j,k, x p j,k [Φ](x j,y k )=s x j,k, qj,k [Φ](x j,y k ) = Φ j,k+ +Φ j,k, y qj,k [Φ](x j,y k )=s y j,k. Substituting this into (34) yields (35) so that L j,k [Φ] = A x A y Φ j,k H(s x j,k,sy j,k ) Φ n+ j,k =( κ)φ n j,k + κl j,k[φ n ].

10 HAILIANG LIU TO MARSHALL SLEMROD ON HIS 7TH BIRTHDAY WITH FRIENDSHIP AND APPRECIATION. 4.. Second order scheme. For the second order scheme, the continuous, piecewise interpolating polynomial is given by p j,k [Φ](x, y) = Φ j,k + s x j,k (x x j )+ (sx j,k ) (x x j )(x x j+ ), qj,k [Φ](x, y) = Φ j,k + s y j,k (y y j )+ (sy j,k ) (y y k )(y y k+ ), where the approximations to the second derivative are selected using the ENO interpolation technique, s x (s x j+,k s x j,k ) j,k = M, sx j,k sx j,k, x x s y (s y j,k ) j,k+ = M sy j,k, sy j,k sy j,k, y y where M{a, b} = a if a b, M{a, b} = b otherwise. We then obtain so that p j,k [Φ](x j,y k ) = Φ j+,k +Φ j,k q j,k [Φ](x j,y k ) = Φ j,k+ +Φ j,k (sx j,k ) ( x), x p j,k [Φ](x j,y k )=s x j,k, (sy j,k ) ( y), y qj,k [Φ](x j,y k )=s y j,k. L j,k [Φ] = A x A y Φ j,k (sx j,k ) Combined with a second order Runge-Kutta method gives, ( x) (sy j,k ) ( y) H(s x j,k,sy j,k ). Φ j,k = ( κ)φ n j,k + κl j,k[φ n ], j,k = Φn j,k + κ Φ j,k + κ L j,k[φ ]. Φ n Third order scheme. The third order scheme formulation is based on the following cubic polynomials where p 3 j,k [Φ](x, y) = p j,k [Φ](x, y)+(sx j,k ) (x x j )(x x j+ )(x x ), 6 qj,k 3 [Φ](x, y) = q j,k [Φ](x, y)+(sy j,k ) (y y k )(y y k+ )(y y ), 6 (s x j,k ) = M (s y j,k ) = M (s x j+,k ) (s x j,k ), (sx j,k ) (s x j,k ), x x (s y j,k+ ) (s y j,k ), (sy j,k ) (s y j,k ). y y

11 THE AE METHOD Here x is chosen as the grid point value used in the ENO procedure for (s x j,k ) so that x = x j 3 or x = x j+3. The value y is chosen in a similar way. This gives p 3 j,k [Φ](x j,y k ) = Φ j+,k +Φ j,k x p 3 j,k = s x j,k (sx j,k ) ( x), 6 q 3 j,k [Φ](x j,y k ) = Φ j,k+ +Φ j,k y qj,k 3 = s y j,k (sy j,k ) ( y). 6 (sx j,k ) (sy j,k ) ( x) (sx j,k ) ( x) (x x ), 6 ( y) (sy j,k ) ( y) (y y ), 6 When combined with the third order Runge-Kutta method, this gives where L j,k [Φ] = A x A y Φ j,k (sx j,k ) H Φ () j,k = ( κ)φ n j,k + κl j,k[φ n ], Φ () j,k = 3 4 Φn j,k + ( κ)φ() 4 j,k + 4 κl j,k[φ () ], Φ n+ j,k = 3 Φn j,k + ( κ)φ() 3 j,k + 3 κl j,k[φ () ], s x j,k (sx j,k ) 6 ( x) (sy j,k ) ( y) (sx j,k ) 6 ( x),s y j,k (sy j,k ) ( y). 6 ( x) (x x ) (sy j,k ) ( y) (y y ) 6 We remark that this scheme can be easily derived for higher orders using an ENO procedure for derivative approximations. 5. Discontinuous Galerkin schemes We begin to formulate discontinuous Galerkin schemes in one dimensional setting. Let the spatial domain be divided to form a uniform grid {x j },with x = x j+ x j. We denote I j =(x j,x j+ ) for j =,...,N withi =(x,x ) and I N =(x N,x N ). Centered at each grid {x j }, the numerical approximation is a polynomial Φ Ij =Φ j (x) P k,wherep k denotes a linear space of all polynomials of degree at most k: P k := {p p(x) Ij = a i (x x j ) i, a i R}. i k Note dim(p k )=k +. We denote v(x ± )=lim ± v(x + ), and v j ± is [v] j = v(x + j ) v(x j ), and the average {v} j = (v+ j + v j ). = v(x ± j ). The jump at x j 5.. The Hamilton-Jacobi equation. We start with the one-dimensional Hamilton-Jacobi equation of the form t φ + H(x, φ, φ x )=. Again the building base is the following AE formulation (36) t u + H(x, u, v x )= (v u).

12 HAILIANG LIU TO MARSHALL SLEMROD ON HIS 7TH BIRTHDAY WITH FRIENDSHIP AND APPRECIATION. Integrating (36) over I j against the test function η P k (I j ), we obtain the semi-discrete AEDG scheme I j ( t Φ j + H(x, Φ j, x Φ SN j ))ηdx + λ[φ SN j ] η(x j )= xj Ij ΦSN j ηdx (37) I j Φ j ηdx, where λ = H p (x j, Φ j (x j ), x Φ j (x j )) and Φ SN j are sampled from neighboring polynomials Φ j± in the following way: xj xj+ H(x, Φ j, x Φ SN j )ηdx = H(x, Φ j, x Φ j )ηdx + H(x, Φ j, x Φ j+ )ηdx, I j x j x j xj xj+ Φ SN j ηdx = Φ j ηdx + Φ j+ ηdx, I j x j x j [Φ SN =Φ j+ (x + j ) Φ j (x j ). xj j ] To update each grid-centered polynomial element Φ, we write the compact form of the semidiscrete scheme d (38) Φ j ηdx = L[Φ j ;Φ SN j,η](t), dt I j where (39) L[Φ j ;Φ j±,η](t) = (Φ SN j I j Φ j )ηdx H x, Φ j, x Φ SN j ηdx λ[φ SN j ] xj η(x j ). I j We use boundary cells I and I n to incorporate proper boundary condtions. For a computational domain [a, b] withx = a, x N = b and x =(b a)/(n ), the equations in two end cells may be given as (4) (4) xn x x ( t Φ + H(x, Φ, x Φ ))ηdx + λ [Φ] x η(x )= x N ( t Φ N + H(x, Φ N, x Φ N ))ηdx + λ [Φ] x N η(x N )= x x xn (Φ Φ )ηdx, x N (Φ N Φ N )ηdx. If the flow is incoming at x = a, one has to impose a boundary condition φ(a, t) =g (t). As a result, one is required to modify (4) by changing [Φ] to Φ (x + ) g (t); for the outflow case, one may take [Φ] = in (4). Similarly, at x = b, the inflow boundary condition φ(b, t) =g (t) can be incorporated in (4) by replacing [Φ] by g (t) Φ N (x N,t); and for outgoing flow, replacing [Φ] by. The determination of the inflow or outflow of the boundary condition may be obtained by checking the sign of the characteristic speed x H(x, φ, p). In the case of periodic boundary conditions, [Φ] can be computed as Φ (x + ) Φ N (x N ) at x = x,x N, and Φ N (x) is regarded to be identical to Φ (x). The fully discrete scheme follows from applying an appropriate Runge-Kutta solver to (38). We summarize the algorithm as follows. Algorithm:. Initialization: in any cell I j, compute the initial data by the local L projection (4) (Φ φ )ηdx =, η P k (I j ). I j. Alternating evaluation : take polynomials Φ j± (x) =Φ Ij±, and then sample in I j to get L[Φ j ;Φ j±,η] as defined in (39).

13 THE AE METHOD 3 3. Evolution: obtain Φ n+ from Φ n by some Runge-Kutta type procedure to solve the ODE system (38). In the AEDG schemes, is chosen such that the stability condition, x t Q max H p (x, φ, p), is satisfied. The choice of Q depends on the order of the scheme. The semi-discrete AEDG scheme is shown to be L -stable for linear Hamiltonian. In particular, when H = αp, α = const., we have the following (see [4]). Theorem 5.. Let Φ be computed from the AEDG scheme (38) for the Hamilton-Jacobi equation t φ + H( x φ)=with linear Hamiltonian H(p) =αp and periodic boundary conditions. Then d dt b a Φ dx = b a (u v) dx. By similar procedures one can construct AEDG schemes for multi-dimensional H-J equations: t φ + H( x φ)=, x := (x,,x d ) R d, based on the AE formulation (3). Let {x α } be distributed grids in R d, and I α be a hypercube centered at x α with vertices at x α±. Centered at each grid {x α }, the numerical approximation is a polynomial Φ Iα =Φ α (x) P r,wherep r denotes a linear space of all polynomials of degree at most r in all x i : P r := {p p(x) Iα = a β (x x α ) β, i d, a β R}. β i r Note dim(p r )=(r + ) d. Integrating the AE system (3) over I α against the test function η P r, we obtain the semi-discrete AEDG scheme I α ( t Φ α + H( x Φ SN α ))ηdx + I α Φ α ηdx = (43) I α Φ SN α ηdx B, B = d (44) j= I α H j ( Φ SN α )[Φ SN α ]η(x)δ(x j x j α j )dx, where Φ SN α are sampled from neighboring polynomials, and H j = pj H(p). The choice of Φ SN α is not unique, and we shall take Φ SN α span{φ α±ej } d j=. We next illustrate our options in the two dimensional case. For α =(i, j), the terms involving neighboring polynomials are as follows: xi+ yj+ x i x i y j H( x Φ SN, y Φ SN )ηdydx = xi+ yj y j xi+ yj x i xi yj+ x i y j xi yj x i H(x, y, Φ i,j, x Φ i+,j, y Φ i,j+ )η(x, y)dydx y j H( x Φ i+,j, y Φ i,j )η(x, y)dydx H( x Φ i,j, y Φ i,j+ )η(x, y)dydx y j H( x Φ i,j, y Φ i,j )η(x, y)dydx.

14 4 HAILIANG LIU TO MARSHALL SLEMROD ON HIS 7TH BIRTHDAY WITH FRIENDSHIP AND APPRECIATION. An average of two neighboring polynomials Φ i±,j and Φ i,j± will be used to evaluate I α Φ SN ηdx, that is Φ SN ηdx = xi+ yj+ (Φ i+,j +Φ i,j+ )η(x, y)dydx I α x i y j xi+ yj x i xi yj+ x i y j xi yj x i The B term in (44) can then be computed as B = + yj+ y j xi+ y j (Φ i+,j +Φ i,j )η(x, y)dydx (Φ i,j +Φ i,j+ )η(x, y)dydx y j (Φ i,j +Φ i,j )η(x, y)dydx. H ( x Φ i,j, y Φ i,j ) Φ i+,j (x + i,y) Φ i,j(x i,y) η(x i,y)dy x i H ( x Φ i,j, y Φ i,j ) Φ i,j+ (x, y + j ) Φ i,j (x, y j ) η(x, y j )dx. For the boundaries, we first consider the side boundary along x = x.then x yj+ x y j H( x Φ SN, y Φ SN )ηdydx = x yj+ x + y j x yj x H( x Φ,j, y Φ,j+ )η(x, y)dydx y j H( x Φ,j, y Φ,j )η(x, y)dydx, and x yj+ x y j Φ SN ηdydx = x yj+ x y j x yj + x (Φ,j +Φ,j+ )η(x, y)dydx y j (Φ,j +Φ,j )η(x, y)dydx, and B = + yj+ y j x H ( x Φ,j, y Φ,j ) Φ,j (x +,y) Φ,j(x,y) η(x,y)dy x H ( x Φ,j, y Φ,j ) Φ,j+ (x, y + j ) Φ,j (x, y j ) η(x, y j )dx. In the above, Φ,j (x,y) may be taken different ways: the given boundary data at x for inflow boundary; Φ,j (x,y) for outgoing flow; and Φ Nx,j(x N x,y) for periodic boundary conditions. Similar computations are made along the other side boundaries x = x Nx,y = y,y = y Ny. For corner cells, we illustrate using the southwest corner (x,y ): and x y x y H( x Φ SN, y Φ SN )ηdydx = x y x y Φ SN ηdydx = x y x x y x y y H( x Φ,, y Φ, )η(x, y)dydx, (Φ, +Φ, )η(x, y)dydx,

15 and B = + y y x x THE AE METHOD 5 H ( x Φ,, y Φ, ) Φ, (x +,y) Φ,(x,y) η(x,y)dy H ( x Φ,, y Φ, ) Φ, (x, y + ) Φ,(x, y ) η(x, y )dx. Similar computations can be made for the other corners (x,y Ny ), (x Nx,y ), (x Nx,y Ny ). Boundary conditions are incorporated in the following ways: boundary data inflow, boundary data inflow, Φ, (x,y)= Φ, (x +,y) outflow, Φ, (x, y Φ Nx,(x )= Φ, (x, y + ) outflow, N x,y) periodic, Φ,Ny (x,y N y ) periodic. The Runge-Kutta method can be used for time discretization with matching accuracy. 5.. Hyperbolic conservation laws. The AEDG method designed in [4] for Hamilton-Jacobi equations can be applied without any difficulty to conservation laws. We only outline the main idea and the scheme formulation. Again the AEDG for one dimensional conservation laws of the form t u + x f(u) = can be derived based on the following AE formulation (45) t u + x f(v) = (v u). Integrating the AE system (45) over I j against the test function η P k (I j ), we obtain the semi-discrete AEDG scheme I j ( t Φ j + x f(φ SN j ))ηdx +[f(φ SN j )] η(x j )= xj Ij ΦSN j ηdx (46) I j Φ j ηdx, where Φ SN j are sampled from neighboring polynomials Φ j± in the following way: xj xj+ x f(φ SN j )ηdx = x f(φ j )ηdx + x f(φ j+ )ηdx, I j x j x j xj xj+ Φ SN j ηdx = Φ j ηdx + Φ j+ ηdx, I j x j x j [f(φ SN j )] = f(φ j+ (x + j )) f(φ j (x j )). xj To update each grid-centered polynomial element Φ j (x), we write the compact form of the semi-discrete scheme d (47) Φ j ηdx = L[Φ j ;Φ SN j,η](t), dt I j where (48) L[Φ j ;Φ j±,η](t) = (Φ SN j I j Φ j )ηdx x f Φ SN j ηdx [f(φ SN j )] xj η(x j ). I j Boundary conditions are incorporated through equations in two end cells. For a computational domain [a, b] withx = a, x N = b and x =(b a)/(n ), the equations in two end cells are given by (49) (5) xn x x ( t Φ + x f(φ ))ηdx + [f(φ)] x η(x )= x N ( t Φ N + x f(φ N ))ηdx + [f(φ)] x N η(x N )= x x xn (Φ Φ )ηdx, x N (Φ N Φ N )ηdx.

16 6 HAILIANG LIU TO MARSHALL SLEMROD ON HIS 7TH BIRTHDAY WITH FRIENDSHIP AND APPRECIATION. If the flow is incoming at x = a, one needs to impose the given boundary condition φ(a, t) = g (t). As a consequence, one is required to modify (49) by changing [f(φ)] to f(φ (x + )) f(g (t)); for the outflow case, one may take [f(φ)] = in (49). Similarly, at x = b, the inflow boundary condition φ(b, t) =g (t) can be incorporated in (5) by replacing [f(φ)] by f(g (t)) f(φ N (x N,t)); and for outgoing flow, replacing [f(φ)] by. The determination of the inflow or outflow of the boundary condition may be obtained by checking the sign of the characteristic speed f (u). In the case of periodic boundary conditions, [f(φ)] can be computed as f(φ (x + )) f(φ N (x N )) at x = x,x N, and Φ N (x) is regarded to be identical to Φ (x). The fully discrete scheme follows from applying an appropriate Runge-Kutta solver to (47). We summarize the algorithm as follows. Algorithm:. Initialization: in any cell I j, compute the initial data by the local L projection (5) (Φ φ )ηdx =, η P k (I j ). I j. Alternating evaluation : take polynomials Φ j± (x) =Φ Ij±, and then sample in I j to get L[Φ j ;Φ j±,η] as defined in (48). 3. Evolution: obtain Φ n+ from Φ n by some Runge-Kutta type procedure to solve the ODE system (47). In the AEDG schemes, is chosen such that the stability condition, x t Q max f (u), is satisfied. The choice of Q depends on the order of the scheme. By a similar procedure one can construct AEDG schemes for multi-dimensional hyperbolic conservation laws: t φ + x f(φ) =, x R d. Again, let {x α } be distributed grids in R d, and I α be a hypercube centered at x α with vertices at x α±. Centered at each grid {x α }, the numerical approximation is a polynomial Φ Iα =Φ α (x) P r,wherep r denotes a linear space of all polynomials of degree at most r in all x i : P r := {p p(x) Iα = a β (x x α ) β, i d, a β R}. β i r Integration of the AE system (3) over I α against the test function η P r, we obtain the semi-discrete AEDG scheme I α ( t Φ α + x f(φ SN α ))ηdx + I α Φ α ηdx = (5) I α Φ SN α ηdx B, B = d (53) j= I α [f(φ SN α )]η(x)δ(x j x j α j )dx, where Φ SN α are sampled from neighboring polynomials. The choice of Φ SN α is not unique, and it is convenient to take the polynomials from the closest neighbors, Φ SN α span{φ α±ej } d j=. The way of sampling from neighboring polynomials for Hamilton-Jacobi equations can be applied well to the system of hyperbolic conservation laws.

17 THE AE METHOD 7 6. Sample numerical experiments The implementation of the AE algorithm began in [4], and further discussed in [4, 4, 43]. Here sample numerical results are taken from [4, 4, 43], respectively. In [4] we use some model problems of hyperbolic conservation laws to numerically test the first, second, and third order AE schemes as illustrated in section. Accuracy tests are based on scalar conservation laws such as the Burgers equation, the nonlinear Buckley-Leverett problem, and multi-dimensional linear transport problem. For shock-capturing tests using the Euler equations of polytropic gas we compared the results with different initial data, including the Lax initial data, the Sod initial data, the Osher-Shu initial data, and the WoodwardColella initial data. In all cases the resolution of the high order finite volume AE schemes is in good agreement with the established results in literature. One example is the explosion problem for the D Euler equation: φ t + f(φ) x + g(φ) y =, φ =(ρ, ρu, ρv, E) T, p =(γ ) E ρ (u + v ), f(φ) =(ρu, ρu + p, ρuv, u(e + p)) T, g(φ) =(ρv, ρuv, ρv + p, v(e + p)) T, where γ =.4. This example chosen from [6] consists of a high density and high pressure region inside a bubble of radius.4 centered at the origin and a low density and pressure region outside the bubble which causes the explosion. The initial data is (p, ρ, u, v) () = (,,, ) x + y <.6, (.,.5,, ) x + y.6. The solution is computed at time T =.5. In this experiment, constant extension boundary condition on all the four walls is used. CFL number used is.4 and time step is taken as t =.95. In this test, the solution exhibits a circular shock and circular contact discontinuity moving away from the center of the circle and circular rarefaction wave moving in the opposite direction, a complex wave pattern emerging as time evolves. We compute this D bubble explosion solution by second order AE scheme. Presented in [4] are extensive numerical tests on both accuracy and capacity of high order finite difference AE schemes for solving Hamilton-Jacobi equations. For accuracy test, we take the example with a non-convex Hamiltonian: subject to initial data t φ cos( x φ + y φ + ) =, π(x + y) φ(x, y, ) = cos, on the computation domain [, ] with periodic boundary conditions. At time t =.5 π,the solution is still smooth and we test the order of accuracy for third order schemes using the dimension-by-dimension approach. The results in Table show the desired order of accuracy in all norms. For capacity of the scheme to capture the singularity, we test the Eikonal equation t φ + x φ + y φ +=, with smooth initial data φ(x, y, ) = (cos(πx) ) (cos(πy) ), 4

18 8 HAILIANG LIU TO MARSHALL SLEMROD ON HIS 7TH BIRTHDAY WITH FRIENDSHIP AND APPRECIATION (a) Density contour (b) Density ρ (c) Pressure contour (d) Pressure p Figure. Isolines of density and pressure to the explosion problem (left). Comparison between the two dimensional solutions and the one dimensional radial solutions (right). nd order AE scheme on [, ] [, ], T =.5, t =.95, x = y = /4. Table. L, L, and L comparison for the non-convex Hamiltonian at time t =.5 using 3rd order dimension-by-dimension AE scheme. π N Scheme AE L error L order L error L order L error L order 4 AE AE AE AE AE and computational domain [, ] and periodic boundary conditions. The results at time t =.6 are shown in Figure. The AE scheme provides high resolution in the formation of the singularity.

19 y THE AE METHOD x.8..4 y x (a) AE scheme, x = y = 8. (b) AE scheme, x = y = 8. Figure. Surface and contour plots at time t =.6 for the D Eikonal equation. In [43], we tested both optimal accuracy and capacity of the AEDG algorithm for solving several time-dependent Hamilton-Jacobi equations. We here take one example relating to controlling optimal cost determination from [37, ] with a nonsmooth Hamiltonian: t Φ+sin(y) x Φ+(sin(x) + sign( y Φ)) y Φ sin (y) + cos(x) =, with initial data Φ(x, y, ) =, on the computation domain [ π, π] with periodic boundary conditions. The results for the numerical solution and optical control sign( t φ) are shown in Figure 3 using P polynomials and are in agreement with those found in [37, ] x 3 y 3 y 3 x (a) Numerical Solution (b) Optimal Control Figure 3. Numerical solution and optimal control plots at time t = with N x = N y = 8 using P polynomials. 7. Concluding remarks and future work In this paper, we present a brief survey of the current state-of-the-art of AE methods for first order partial differential equations. We demonstrate the main ideas of the algorithm through two canonical model equations: hyperbolic conservation laws and Hamilton-Jacobi equations. We include some discussions of the properties and extensions of the schemes. Central solvers

20 HAILIANG LIU TO MARSHALL SLEMROD ON HIS 7TH BIRTHDAY WITH FRIENDSHIP AND APPRECIATION. have recently gained growing attention in the field of conservation laws and Hamilton-Jacobi equations because of the guaranteed accuracy and reliable simulation results they provide. For both conservative and non-conservative PDEs we are able to apply the same discontinuous Galerkin framework upon the AE formulation. From the stand point of algorithm design and development, it will be interesting to explore ways to utilize fully the freedom of the AEDG framework such as hp-adaptivity. For practical purposes, we will develop AE models and solvers to simulate multi-phases fluids and problems involving multi-scales in the future. Extensions to both stationary Hamilton-Jacobi equations and the convection-diffusion equations using section 5. are currently underway. In recent years, numerical solutions of fully nonlinear second order PDEs have attracted a great deal of attention from the numerical PDE and scientific communities, we refer to [5] for a recent review on this subject and references therein. Extension of our AE methods to second order fully nonlinear PDEs will also be an important component of our future research. Acknowledgments This research was partially supported by the National Science Foundation under grant DMS and the KI-Net research network. References [] F. Bianco, G. Puppo, and G. Russo. High-order central schemes for hyperbolic systems of conservation laws. SIAM J. Sci. Comput., ():94 3 (electronic), 999. [] S. Bryson, A. Kurganov, D. Levy, and G. Petrova. Semi-discrete central-upwind schemes with reduced dissipation for Hamilton-Jacobi equations. IMA J. Numer. Anal., 5():3 38,5. [3] S. Bryson and D. Levy. High-order central WENO schemes for multidimensional Hamilton-Jacobi equations. SIAM J. Numer. Anal., 4(4): (electronic), 3. [4] S. Bryson and D. Levy. High-order semi-discrete central-upwind schemes for multi-dimensional Hamilton- Jacobi equations. J. Comput. Phys., 89():63 87, 3. [5] M.G. Crandall and L.C. Evans and P.L. Lions. Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 8: 487 5, 984. [6] Y. Cheng and C.-W. Shu. A discontinuous Galerkin finite element method for directly solving the Hamilton- Jacobi equations. J. Comput. Phys., 3():398 45, 7. [7] B. Cockburn and C.-W. Shu. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput., 6(3):73 6,. [8] P. Colella and P. R. Woodward. The piecewise parabolic method (PPM) for gas-dynamical simulations. J. Comput. Phys., 54():74,984. [9] M. G. Crandall and P.-L. Lions. Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc., 77(): 4, 983. [] B. Cockburn and S. Hou and C.-W. Shu. The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case. Math. Comput. 54 (9): , 99. [] B. Cockburn and S.Y. Lin and C.-W. Shu. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. III. One-dimensional systems. J. Comput. Phys. 84 (): 9 3, 989. [] B. Cockburn and C.-W. Shu. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Math. Comput. 5 (86): 4 435, 989. [3] B. Cockburn and C.-W. Shu. The Runge-Kutta local projection P discontinuous Galerkin finite element method for scalar conservation laws. RAIRO model. Math. Anal. Numer 5 (3): , 99. [4] B. Cockburn and C.-W. Shu. The Runge-Kutta discontinuous Galerkin method for conservation laws. V. Multidimensional systems. J. Comput. Phys. 4 (): 99 4, 998. [5] X.-B. Feng and R. Glowinski and M. Neilan. Recent developments in numerical methods for fully nonlinear second order partial differential equations. SIAM Review, 55(): 5 67, 3. [6] K. O. Friedrichs. Symmetric hyperbolic linear differential equations. Comm. Pure Appl. Math., 7:345 39, 954.

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