A Discontinuous Galerkin Moving Mesh Method for Hamilton-Jacobi Equations

Transcription

1 Int. Conference on Boundary and Interior Layers BAIL 6 G. Lube, G. Rapin Eds c University of Göttingen, Germany, 6 A Discontinuous Galerkin Moving Mesh Method for Hamilton-Jacobi Equations. Introduction J.A Mackenzie and A. Nicola Department of Mathematics, University of Strathclyde, 6 Richmond St, Glasgow. jam@maths.strath.ac.uk In this paper we consider the adaptive numerical solution of Hamilton-Jacobi HJ equations φ t + Hφ,...,φ d =, φ, = φ, where =,..., d IR d, t >. HJ equations arise in many practical areas such as differential games, mathematical finance, image enhancement and front propagation. It is well known that solutions of are Lipschitz continuous but derivatives can become discontinuous even if the initial data is smooth. There is a close relation between HJ equations and hyperbolic conservation laws. With this in mind, it not surprising to find that many of the numerical methods used to solve HJ equations are motivated by conservative finite difference or finite volume methods for conservation laws. An increasingly popular approach to solve hyperbolic conservation laws is the discontinuous Galerkin DG finite element method. Recently, Hu and Shu [] proposed a DG method to solve HJ equations by first rewriting as a system of conservation laws w i t + Hw i =, i =,...,d, w, = φ, where w = φ. The usual DG formulation would be obtained if w belonged to a space of piecewise polynomials. However, we note that w i, i =,...,d are not independent due to the restriction that w = φ. In [] a least squares procedure was used to enforce this condition whereas in [4] a suitable basis was used such that w h = φ h The aim of this paper is to consider the use of the DG method of Hu and Shu [] to solve HJ equation using a moving mesh method based on the solution of moving mesh PDEs []. The governing equation are transformed to include the effect of the movement of the mesh and this is done in such a way that the conservation properties of the original equation are not lost. The adaptive mesh is driven by a monitor function which is shown to be non-singular in the presence of solution discontinuities.. DG discretisation of Hamilton-Jacobi equations... HJ equation in D Let us consider the Hamilton-Jacobi equation in one dimension { φt + Hφ =,,t a,b,t] φ, = φ, with appropriate boundary conditions. If u = φ and we differentiate 3 with respect to then u satisfies the hyperbolic conservation law { ut + Hu =,,t a,b,t] 4 u, = u. 3

2 We assume that the physical domain Ω p = [a,b] is the image of a computational domain Ω c = [,] which is obtained via the time-dependent mapping = ξ,t. We will consider a discretisation of the conservative form in computational coordinates { ξ u + Hu ẋu ξ =, ξ,t,,t], 5 uξ, = u ξ. The aim is to discretise 5 in space using a DG method.... The DG discretisation { } N For each partition ξ j+ of Ω c, we denote j= I j = ξ j,ξ j+, j = ξ j+ ξ j, j =,...,N. We use a uniform mesh to cover the computational domain such that h = ξ j+ ξ j = N, ξ j = ξ j+ + ξ j, j =,...,N. 6 If we multiply 5 by an arbitrary smooth function v, then integrating by parts over the interval I j we obtain ξ uv dξ Hu,ẋvξ dξ + Hu,ẋv =, 7 I j I Ij j where and Hu, ẋv = Huξ j+ Ij Hu,ẋ = Hu ẋu, 8,t,ẋξ j+ vξ j+ Huξ j,t,ẋξ j vξ j. 9 We will approimate the eact solution u of 7 and the smooth function v locally by polynomials of degree at most k. The numerical approimations u h and v h belong to the finite dimensional space } Vh {v k = : v Ij P k I j, j =,...,N, where P k I j denotes the space of polynomials on I j of degree at most k. In the discontinuous Galerkin numerical method u h and v h are discontinuous at the points ξ j+, j =,...,N, and the question is how to evaluate their values in 9. We set v h j+ = lim v h ξ, and v + ξ ξ h = lim v j h ξ. j+ ξ ξ + j Furthermore, we replace the nonlinear flu function Hu h,ẋ defined in 8 by a numerical flu function that depends on the values of u h from the left and right at the point,t, that is Hu h,ẋ ξj+ t = H j+ In this paper we have used the local La-Friedrichs flu Hp,q,ẋ = ξ j+ u h ξ,t,u j+ h ξ +,t,ẋ j+ ξj+ t. Hp,ẋ + Hq,ẋ cq p, 3

3 where c = ma minp,q u map,q H u. The discontinuous Galerkin space approimation is given as the solution of the weak formulation ξ u h v h dξ Huh,ẋv h ξ dξ + H j+ v h ξj+ H j v + h ξj =, 4 I j I j and the initial condition is given by the projection u h ξ,v h ξ dξ = u ξv h ξ dξ, j =,...,N, 5 I j I j for all v h V h.... Temporal integration The discontinuous Galerkin discretisation gives rise to a system of ODEs which can be written as { d dt ξu h = L h u h, in,t] 6 u h = u h, where u h is the initial approimation for the vector of the degrees of freedom u h, and ξ depends on the meshes at time level t n and t n+ as defined earlier. To integrate 6 numerically we use the second-order TVD Runge-Kutta method n+ ξ u h = n ξ un h + t Lun h, n+ ξ u n+ h = n ξ un h + n+ ξ u h + 7 t Lu h.... Recovery of φ h If φ h = u h, then φ h will be determined on each interval I j up to a constant. Following [], the constant can be retrieved in two ways:. require that I j φ h t + Hu h d, j =,...,N. 8. use 8 to update only the leftmost element I and then since φ h = u h we have..3. HJ equations in D In two-dimensions we have φ h j,t = φ h,t + j u h,t d. 9 φ t + Hφ,φ y =,,y Ω p, t >, where Ω p lr is the physical domain. To generate an adaptive mesh we have seen earlier that it is useful to regard the physical domain Ω p as the image of a computational domain Ω c under the invertible maps = ξ,η,t, y = yξ,η,t, and ξ = ξ,y,t, η = η,y,t, 3

4 where =, y and ξ = ξ, η are the physical and computational coordinates, respectively. A mesh covering Ω p is obtained by applying the mapping given by to a partitioning of Ω c. Given that the mesh can move then it is necessary to epress the Eulerian -fied temporal derivative in in terms of the Lagrangian derivative along the trajectory of the moving mesh. If a dot denotes differentiation with respect to time with ξ,η fied then becomes φ ẋφ ẏφ y + Hφ,φ y =, where ẋ,ẏ represents the mesh velocity. If we let u = φ ξ and v = φ η, then by differentiating with respect to ξ and η we arrive at the system u t + H ξ u,v,ẋ,ẏ =, 3 where v t + H η u,v,ẋ,ẏ =, 4 Hu,v,ẋ,ẏ = Hξ u + η v,ξ y u + η y v ẋξ u + η v ẏξ y u + η y v. 5 Using the vectorial notation u = u v H, F =, F = H we can rewrite 3 and 4 as the system of conservation laws u t + [F u] ξ + [F u] η =. 6 We apply a discontinuous Galerkin method to solve this coupled system 6 for u,v = ξ φ, where the discretisation takes place in the computational domain Ω c. A recovery procedure will then be used to obtain an approimation of φ. To obtain a weak formulation we take the inner product of 6 with a test function v h V h, integrate over K T h and replace u by its approimation u h V h. That is d dt K u h ξ,t v h ξ dξ + and using integration by parts we obtain d u h ξ,t v h ξ dξ dt K + e K e K, [F ξ v h + F η v h ] dξ =, 7 { } n e,k F v h ξ intk + n e,k F v h ξ intk ds [F v h ξ + F v h η ] dξ =, K where n e,k = [n e,k,n e,k ] is the unit normal to the edge e of the element K, intk denotes the value taken from the within the element K, and etk denotes the value taken from the eterior of the element K. Net we define the normal flu F e,k u,ẋ,ẏ = n e,k F + n e,k F = [ ne,k H n e,k H 8 ]. 9 Note that F e,k is not well defined on the element edge as u h is discontinuous there. Therefore, we replace it by the numerical flu function F e,k u h ξ intk,u h ξ etk,ẋ,ẏ. 3 4

5 Again we will use the La-Friedrichs flu F e,k a,b,ẋ,ẏ = [F e,ka + F e,k b + α e,k a b], 3 where F α e,k = γρ n e,k u + n F e,k, 3 u and γ and ρ is the spectral radius of the normal Jacobian matri. We now discuss the basis used when u h is a discontinuous bilinear function. Following Hu and Shu [] we assume that the approimate solution of the Hamilton-Jacobi equation φ h is piecewise discontinuous and quadratic. Therefore, in the cell K ij we assume where φ h t = φt + φ ξ tµ i + φ η tν j + φ ξη tµ i ν j +φ ξξ t µ i + φ ηη t νj 3, 33 3 µ i ξ = ξ ξ i ξ, and ν j η = η η j. η The gradient in computational space ξ φ h therefore takes the form [ ξ ξ φ h = φ ξt + ξ φ ξηtν j + 4 ξ φ ] ξξtµ i η φ ηt + η φ ξηtµ i + 4 η φ, 34 ηηtν j and hence there are five unknowns that determine ξ φ h. A suitable basis for V = {v,v,v 3,v 4,v 5 : v K = ξ φ, φ P K, K T h } is given by b = [ ] [, b = ] [ µi, b 3 = ] [, b 4 = ν j ] [ νj, b 5 = ξ µ i η ]. 35 The local basis functions 35 can easily be shown to be orthogonal and hence the mass matri is diagonal. 3. Moving mesh equations Let us consider =, T to be a point in the physical domain Ω p and ξ = ξ,ξ T to be a point in the computational domain Ω c. The mapping from the computational domain to the physical domain is assumed to satisfy the moving mesh PDE [] τ t = a i,i + b i i= a i G a j ξ i ξ j i,j= a i G a j, 36 ξ j ξ i i,j= where a i = ξ i and a i,j = a i G a j, b i = i,j= a i G a j. 37 ξ j 5

6 Dirichlet boundary conditions for the above system are obtained by solving a one-dimensional MMPDE. If p Ω p and c Ω c denote the physical and computational boundary segments with arc-lengths l and l c respectively, then the mesh on p is the solution of τ s t = M + M ζ ζ M s ζ, ζ,l c, 38 with s = and sl c = l. Here M is the one-dimensional projection of the two-dimensional monitor function along the boundary. That is, if t is a unit tangent vector along the boundary then Ms,t = t T Gt Choice of monitor function The choice of a suitable adaptivity criterion for problems which develop shocks is highly nontrivial. It is important that the mesh points move smoothly towards regions where discontinuities eist so that the O error at the shock is localised within one or two mesh elements. For robustness, it is also important that the rate of convergence of the method is not severely impaired by the non-uniformity of the moving mesh. To obtain a bounded monitor function we consider the choice [3] M,t = + u, α = α β ma u, 39 and β is a user-chosen parameter. It is clear that this monitor function satisfies the bounds M,t + β. If the mesh is found by equidistribution of the monitor function then if M ecept in the cell containing then + β min b a + + β N min. 4 Therefore, the minimum mesh spacing is min b a N + β 4 which is approimately a factor of + β smaller than that using a uniform mesh. For two dimensional problems the monitor function used is of Winslow type where [ ] w G = w and w = + u α, where α = β ma Ωp u. This monitor function behaves in a similar fashion to the one-dimensional monitor in the presence of solution discontinuities. 4. Numerical eperiments 4... Eample The first eample considered is Burgers equation φ t + φ + =, < <, t > φ, = cosπ.85, with periodic boundary conditions. 4 6

7 u h u h φ h.4 φ h Figure : DG numerical solution and eact solution + obtained on a uniform stationary mesh left, and on a moving mesh right at time t = 3/π Eample We consider the two-dimensional Burgers equation with periodic boundary conditions. 5. Conclusions φ t + + φ + φ y =,,y, 43 φ,y, = cosπ + y/, 44 We have presented a DG method for the solution of nonlinear Hamilton-Jacobi equations using an adaptive moving mesh refinement strategy. A suitable mesh refinement criterion was used which does not become singular when the solution become non-smooth. Numerical eperiments in one and two dimensions show that method works well and efficiently delivers high resolution solutions using relatively coarse meshes. References [] C. Hu and C.W. Shu. A Discontinuous Galerkin Finite Element Method for Hamilton- Jacobi Equations. SIAM Journal on Scientific Computing, :666 69,

8 .5.5 φ.5 φ.5.5 y.5 y Figure : Approimations of φ top and φ bottom at t =.5/π using uniform mesh left and using moving mesh right, with N = y.5 y Figure 3: Adapted meshes at t =.5/π left, and t =.5/π right obtained with N = 4 for Eample. [] W. Huang. Practical aspects of formulation and solution of moving mesh partial differential equations. Journal of Computational Physics, 7: ,. [3] J.M. Stockie, J.A. Mackenzie, and R.D. Russell. A Moving Mesh Method for One- Dimensional Hyperbolic Conservation Laws. SIAM Journal on Scientific Computing, 5:79 83,. [4] F. Li and C.W. Shu. Reinterpretation and simplified implementation of a discontinuous Galerkin method for Hamilton-Jacobi equations. Applied Mathematics Letters, 8:4 9, 5. 8