A Supra-Convergent Finite Difference Scheme for the Poisson and Heat Equations on Irregular Domains and Non-Graded Adaptive Cartesian Grids

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1 Journal of Scientific Computing, Vol. 3, Nos. /, May 7 ( 7) DOI:.7/s A Supra-Convergent Finite Difference Scheme for the Poisson and Heat Equations on Irregular Domains and Non-Graded Adaptive Cartesian Grids Han Chen, Chohong Min, and Frédéric Gibou 3 Received March 3, 6; accepted (in revised form) November 8, 6; Published online March 3, 7 We present finite difference schemes for solving the variable coefficient Poisson and heat equations on irregular domains with Dirichlet boundary conditions. The computational domain is discretized with non-graded Cartesian grids, i.e., grids for which the difference in size between two adjacent cells is not constrained. Refinement criteria is based on proximity to the irregular interface such that cells with the finest resolution is placed on the interface. We sample the solution at the cell vertices (nodes) and use quadtree (in D) or octree (in 3D) data structures as efficient means to represent the grids. The boundary of the irregular domain is represented by the zero level set of a signed distance function. For cells cut by the interface, the location of the intersection point is found by a quadratic fitting of the signed distance function, and the Dirichlet boundary value is obtained by quadratic interpolation. Instead of using ghost nodes outside the interface, we use directly this intersection point in the discretization of the variable coefficient Laplacian. These methods can be applied in a dimension-by-dimension fashion, producing schemes that are straightforward to implement. Our method combines the ability of adaptivity on quadtrees/octrees with a quadratic treatment of the Dirichlet boundary condition on the interface. Numerical results in two and three spatial dimensions demonstrate second-order accuracy for both the solution and its gradients in the L and L norms. KEY WORDS: Poisson equation; heat equation; irregular domains; supraconvergence; non-graded adaptive Cartesian grids; quadtrees; octrees. Computer Science Department, University of California, Santa Barbara, CA 936, USA. Department of Mathematics, KyungHee University, 3 7, Seoul, Korea. 3 Mechanical Engineering Department and Computer Science Department, University of California, Santa Barbara, CA 936, USA. fgibou@engineering.ucsb.edu /7/5-9/ 7 Springer Science+Business Media, LLC

2 Chen, Min, and Gibou. INTRODUCTION The Poisson and the heat equations are two of the fundamental equations used in the modeling of diffusion dominated phenomena, ranging from electromagnetism to semi-conductor growth to fluid mechanics. A wide variety of approaches exist for solving the Poisson equation on irregular domains. In [6] Peskin introduced the immersed boundary method, which uses a δ-function that smears out the solution on a thin finite band around the interface. Leveque and Li introduced the immersed interface method [6], which is a second-order accurate numerical method designed to preserve the jump condition at the interface. In [, 4], Mayo et al. proposed a fast method using boundary integral techniques. Based on the Ghost Fluid Method of Fedkiw et al. [8], Liu et al. [8] developed a firstorder accurate symmetric discretization of the variable coefficient Poisson equation in the presence of an irregular interface across which the variable coefficient, the solution and its derivative may have jumps. Gibou et al. [] introduced a second-order accurate symmetric discretization of the variable coefficient Poisson equation in the case where Dirichlet boundary conditions are imposed on the irregular domain. In this case, ghost values across the interface are defined by linear extrapolation allowing standard central differencing formulas to be used when discretizing the Laplacian near the interface. Third and fourth-order accurate schemes were later proposed in Gibou et al. [9] by defining ghost values with higher-order extrapolations. In this case the corresponding linear system is non-symmetric but the method produces accurate results for very coarse grids, making the method extremely efficient. Jomaa and Macaskill [4] showed that in the case where ghost values are defined by linear extrapolations, the error near the boundary is in general large compared to the error found for regular domains. Defining ghost values by a quadratic extrapolation makes the linear system non-symmetric, but leads to errors comparable with those obtained for a regular domain. Many physical problems have different scales and more often than not, only small portions of the computational domain require finer resolution. In this case, uniform grids become inefficient in terms of memory storage and CPU usage, and adaptive mesh strategies are usually adopted. Adaptive mesh strategies for elliptic partial differential equations, such as the Poisson equations, are often associated with the finite element method (see, e.g., [7, 3]). Young et al. [35] introduced a finite element method employing adaptive mesh refinements for second-order variable coefficient elliptic equations using a cut-cell representation of irregular domains. However, adaptive mesh refinement strategies for the finite element method can be complex in the case of irregular domains due to the

3 A Supra-Convergent Finite Difference Scheme organization of the resulting data structure. In addition, special care must be taken for moving boundary problems in order to guarantee good mesh quality (see, e.g., [3]). Cartesian grids have the advantage of easier grid generation over unstructured grids, especially in three spatial dimensions. In [], Johansen and Colella presented a cell-centered finite volume discretization to solve the variable coefficient Poisson equation on irregular domains with Dirichlet boundary conditions. They used a multigrid approach and a block-grid algorithm related to the adaptive mesh refinement scheme of Berger and Oliger [5]. McCorquodale et al. [3] presented a node-centered finite difference approach to solving the variable coefficient Poisson equation with Dirichlet boundary conditions on irregular domains using the block-structured adaptive mesh refinement and the multigrid solver of Almgren et al. [ 4]. Numerical evidence in [3] showed that while both linear and quadratic interpolations produce second-order accurate solutions, only quadratic interpolations (Shortley Weller approximation [3]) yield second-order accurate gradients in the L norm. These methods, however, do not consider non-graded adaptive grids in which the difference in size between two adjacent cells is not constrained. Aftomis et al. [] pointed out that quadtree/octree data structures (see, e.g., [9, 3] for an introduction on quadtree/octree data structures) could be used as an efficient means to represent Cartesian grids. Popinet [7] proposed a second-order non-symmetric numerical method to study the incompressible Navier Stokes equations using an octree data structure to represent the spatial discretization. In this method, a Poisson equation for the pressure is solved to account for the incompressibility condition using a standard projection method. Only graded trees, i.e., trees for which the ratio between two adjacent cell sizes cannot exceed two, were considered in [7]. Non-graded grids generate less nodes leading to efficient methods and allow for more flexible grid generations since the size between adjacent cells is not constrained. In [], Losasso et al. proposed a first-order accurate symmetric discretization of the Poisson equation on non-graded octrees in the context of free surface flows. This work was then extended to second-order accuracy in [9] using the work of Lipnikov et al. [7]. A common criticism of quadtree/octree data structures is the time it takes to traverse the tree from the root to a leaf. This issue can be alleviated by using a uniform grid at the coarsest level and a quadtree/octree in every cell of this uniform grid as demonstrated in [9]. Recently, Min et al. [5] introduced a supra-convergent finite difference scheme for solving the variable coefficient Poisson equation on regular domains using non-graded grids represented by quadtrees/octrees. A hallmark of this method is that it produces second-order accurate

4 Chen, Min, and Gibou solutions with second-order accurate gradients (supra-convergence). In this paper, we extend this method to solve the variable coefficient Poisson and the heat equations on irregular domains. We describe the interface by the set of points where a level set function is zero. We automatically construct non-graded Cartesian grids based on the proximity to the interface, i.e., we impose the finest resolution at the interface. Such grid generations yield straightforward approximations of both the Poisson and the heat equations on non-graded Cartesian meshes. We note, however, that one can in addition refine the mesh in other regions if needed, as shown in Sec. 5. For interior nodes, the variable coefficient Laplacian is discretized as in [5], where the spurious error induced by interpolation at ghost nodes at T-junctions is cancelled out by a weighted combination of the discretizations in other Cartesian directions. For nodes adjacent to the interface, the location of the interface point is found by a quadratic fitting of the signed distance level set function, and the Dirichlet boundary value is obtained either explicitly (if an analytic formula is known) or calculated using a quadratic interpolation using neighboring nodal values in the case where the Dirichlet boundary values are defined at the nodes. The Shortley Weller approximation [3], based on quadratic extrapolation, is then used to discretize the Laplacian. The resulting non-symmetric linear system is solved using the stabilized bi-conjugate gradient (Bi-CGSTAB) method with the incomplete LU preconditioner [8]. In the case of the heat equation, the Crank Nicolson scheme is used with a time step of t = c x, where <c< and x is the size of the smallest cell. Numerical examples demonstrate that our method produces second-order accuracy for the solution as well as its gradients in the L and L norms. We emphasize that second-order accuracy for the gradient is particularly important in Stefantype problems since the overall accuracy of the method is determined by the accuracy of the gradients at the interface. We find that our method on adaptive grids favorably compares to the symmetric solver of Gibou et al. [] on uniform grids.. EQUATIONS.. Poisson Equation Consider a Cartesian computational domain, Ω, with exterior boundary, Ω, and a lower-dimensional irregular interface, Γ, which divides the computational domain into disjoint pieces, the interior region Ω and the exterior region Ω + (see Fig. ). The regions are represented by a level set function φ, taken to be the signed distance function to the interface Γ. That is, Ω is represented by the set of points where φ< and Ω + is

5 A Supra-Convergent Finite Difference Scheme 3 Fig.. Schematic of two distinct regions Ω and Ω +, separated by an interface Γ. represented by the set of points where φ>, whereas Γ is represented by the set of points where φ = The variable coefficient Poisson equation is written as (ρ( x) u( x))= f( x), x Ω, () where =( x, y, z ) is the gradient operator and where ρ( x) is assumed to be continuous on each of the disjoint subdomains, Ω and Ω +, but may be discontinuous across the interface Γ. Furthermore, ρ( x) is assumed to be bounded from below by a positive constant. On Ω, either Dirichlet or Neumann boundary conditions are specified. On the interface Γ, a Dirichlet boundary condition of u Γ = g( x) is specified. Thus, Eq. () decouples into two distinct equations, one on Ω and one on Ω +, and the solutions can be obtained independently... Heat Equation Ignoring the effects of convection, the standard heat equation reads ρc v T t = (k T), () where T is the temperature, ρ the density, c v the specific heat at constant volume, and k is the thermal conductivity. Assuming that ρ and c v are

6 4 Chen, Min, and Gibou constant allows Eq. () to be written as where ˆk = k ρc v. T t = (ˆk T), (3) 3. SPATIAL DISCRETIZATION ON QUADTREES/OCTREES The computational domain Ω is discretized into squares (in D) or cubes (in 3D), and quadtree (in D) or octree (in 3D) data structures are used to represent this discretization. As depicted in Fig. 6 for a D domain, the entire domain is originally associated with the root of the tree, which has a level of zero by definition. Then this discretization proceeds recursively, i.e., each cell can be in turn split into four children which have one more level than their parent cell. A cell with no children is called a leaf. Two cells are called neighbors if they share a common face or part of a face. The interested reader is referred to [9, 3] for more details on quadtree and octree data structures (Fig. ). By definition, a quadtree is graded if the difference between two adjacent cell levels is at most one. In this paper, we sample the solution at the nodes (vertices of cells) and we consider non-graded Cartesian grids, i.e., grids for which the level difference between two adjacent cells is not constrained. In our non-graded Cartesian grids, a node is regular if it is linked to another node by a cell edge in each of the directions. In two Fig.. Discretization of a D domain (left) and its quadtree representation (right). The entire domain corresponds to the root of the tree (level ), and each cell is subdivided into four children, in the order of lower-left, upper-left, lower-right, upper-right. In this example, the tree is not graded since the difference of levels between neighboring cells exceeds one.

7 A Supra-Convergent Finite Difference Scheme 5 Fig. 3. A general configuration in D illustrating the nodes involved in the discretization at a T-junction node v. The broken line between v and the ghost node v 4 is imaginary. Fig. 4. A general configuration in 3D illustrating of the nodes involved in the discretization at v. There is a D T-junction in the x-direction, and a 3D T-junction in the y-direction. The broken lines are imaginary. spatial dimensions, a node is called a T-junction node if it does not have a direct neighboring node in one, and only one, of the x, x, y, and y-directions, as illustrated in Fig. 3. In three spatial dimensions, at most one 3D T-junction and one D T-junction can occur at a node, as depicted in Fig. 4.

8 6 Chen, Min, and Gibou Adaptive grids are usually generated in such a way as to produce a uniform error throughout the entire domain, see, e.g., [34] for adaptive mesh generations based on a posteriori error estimate. For many problems, e.g., Stefan-type problems [6, ], the solution s gradient at the interface drives the interface motion and therefore the accuracy of the method. It is therefore desirable to discretize the computational domain in such a way that the cell size is proportional to the distance to the interface. In this work, we split a cell c if the following test is satisfied: min v V lip diag φ(v) <, (4) where v is a vertex of the current cell c, V the set of all vertices of c, lip the Lipschitz constant associated with φ, and diag is the diagonal length of the current cell. As pointed out by Strain in [33], this is a splitting algorithm that automatically generates non-graded Cartesian grids and guarantees the finest resolution on the interface. In addition, enforcing uniform cells with the smallest size on the interface prevents the occurrence of T-junctions. This greatly simplifies our finite difference discretization in Sec. 4. and allows a straightforward extension of the method of Min et al. [5]. 4. FINITE DIFFERENCE SCHEMES 4.. Variable Coefficient Poisson Equation on Regular Domains We first recall the finite difference discretization from Min et al. [5] for the variable coefficient Poisson Eq. () on regular domains One Spatial Dimension In one spatial dimension, the finite difference discretization on a nonuniform grid at a grid node x i is ( ) u i+ u i ρi+ + ρ i + u i u i ρi + ρ i s i+ s i s i+ + s i = f i + s i+ s i 3 u xxx + O(h ), (5) where s i =x i x i and h=max i s i+. Even though this scheme appears to be second-order accurate only in the case where the grid is uniform, it is in fact second-order accurate (see, e.g., [5, ]).

9 A Supra-Convergent Finite Difference Scheme Two Spatial Dimensions In two spatial dimensions, a node is either a regular node or a T-junction node, i.e., a node with a missing direct neighboring node. We emphasize that the T-junction can only occur at one of the x, +x, y, and +y-directions. Let us consider the finite difference discretization for a T-junction node without a direct right neighboring node as depicted in Fig. 3 noting that the discretization when the T-junction is in the other directions can be derived in a similar manner: denote by u i the solution value at node v i. The ghost value u 4 is calculated by linear interpolation using u 5 and u 6 : u 4 = s 5u 6 + s 6 u 5 s 5 + s 6. (6) The forward difference for ρu x at node v is also linearly interpolated as where (ρu x ) + v = s 5D 6 + s 6 D 5 s 5 + s 6, (7) D 5 = u 5 u ρ5 + ρ, s 4 (8) D 6 = u 6 u ρ6 + ρ. s 4 (9) The discretizations for (ρu x ) x and (ρu y ) y at the node v, along with their Taylor series expansions, are given by ( u u ρ + ρ s = (ρu x ) x + ) + s 6D 5 + s 5 D 6 s 5 + s 6 s + s 4 s 5 s 6 ( ) ρuy + O(h) () (s + s 4 )s y 4 and ( u u ρ + ρ + u ) 3 u ρ3 + ρ = ( ) ρu y + O(h), () s s 3 s + s y 3 respectively, where D 5 and D 6 are given by Eqs. (8) and (9). The spurious term s 5s 6 ( (s +s 4 )s ρuy 4 is due to the T-junction, and can be canceled by )y

10 8 Chen, Min, and Gibou appropriately weighting Eqs. () and () as: ( u u ρ + ρ + s ) 6D 5 + s 5 D 6 s s 5 + s 6 s + s ( 4 u u + ρ + ρ + u ) 3 u ρ3 + ρ s s 3 ( ) s 5 s 6 = f + O(h). () s + s 3 (s + s 4 )s 4 Although this scheme appears to be first-order accurate, it is indeed second-order accurate as in the one spatial dimension. Not surprisingly, this weighted scheme () reduces to the usual five point scheme in the case of a regular node: ( u u ρ+ρ s ( u u + + u 4 u ρ4+ρ ) s 4 ρ+ρ + u 3 u ρ3+ρ s s 3 s +s ) 4 where v 4 is the direct neighboring node of v to the right. s +s 3 =f +O(h), (3) Three Spatial Dimensions In three spatial dimensions, more complex configurations can occur due to the adaptive nature of the Cartesian grid. The most general case is depicted in Fig. 4, where one D T-junction in the x-direction and one 3D T-junction in the y-direction are present. We emphasize that this is the most general case and that no other T-junctions can occur at a single node. In this case two ghost values u 4 and u 5 are linearly interpolated as: and u 4 = u 7s 8 + u 8 s 7 s 8 + s 7 (4) u 5 = u s s + u s s 9 + u 9 s s + u s s 9. (5) (s + s )(s 9 + s ) The backward difference (ρu x ) and (ρu y ) at node v are also linearly interpolated as (ρu x ) = ρ 7 + ρ u7 u s 4 s 8 + ρ 8 + ρ u8 u s 8 + s 7 s 4 s 7 s 8 + s 7 (6)

11 A Supra-Convergent Finite Difference Scheme 9 and (ρu y ) = ρ + ρ + ρ + ρ + ρ + ρ + ρ 9 + ρ u u s 5 u u s 5 u u s 5 u9 u s 5 s s (s + s )(s 9 + s ) s s 9 (s + s )(s 9 + s ) s s 9 (s + s )(s 9 + s ) s s (s + s )(s 9 + s ), (7) respectively. The discretizations for (ρu x ) x, (ρu y ) y, and (ρu z ) z at the node v, along with their Taylor series expansions, are given by ( ρ + ρ ( ρ + ρ and ( ρ3 + ρ u u s u u s ) (ρu x ) ) (ρu y ) s + s 4 = (ρu x ) x + u3 u + ρ ) 6 + ρ u6 u s 3 s 6 s 7 s 8 s 4 (s + s 4 ) (ρu z) z + O(h), (8) = ( ) ρu y s + s y + s 9s 5 s 5 (s + s 5 ) (ρu x) x + s s s 5 (s + s 5 ) (ρu z) z + O(h) (9) s 3 + s 6 = (ρu z ) z + O(h), () respectively, where (ρu x ) and (ρu y ) are given by Eqs. (6) and (7). As in two spatial dimensions, a linear combination of Eqs. (8) () s allows for the cancellation of the spurious errors terms 7 s 8 s 4 (s +s 4 ) (ρu z) z, s 9 s s 5 (s +s 5 ) (ρu s x) x, and s s 5 (s +s 5 ) (ρu z) z, yielding the following scheme: ( ) ρ + ρ u u + (ρu x ) α s s + s ( ) 4 ρ + ρ + u u + (ρu y ) s s + s ( 5 ρ3 + ρ + u3 u + ρ ) 6 + ρ u6 u β = f + O(h) s 3 s 6 s 3 + s 6 ()

12 3 Chen, Min, and Gibou with α = s s s 5 (s + s 5 ), β = s 9s s 5 (s + s 5 ) α s 7 s 8 s 4 (s + s 4 ). () Although this scheme appears to be first-order accurate, it is in fact second-order accurate as it is in the D case. Not surprisingly, this weighted scheme () reduces to the usual seven point scheme in the case of a regular node: ( u u ρ + ρ s ( u u + ρ + ρ s + + u ) 4 u ρ4 + ρ s 4 + u 5 u ρ5 + ρ s 5 ( u6 u ρ6 + ρ + u 3 u ρ3 + ρ s 6 s 3 s + s ) 4 s + s ) 5 s 6 + s 3 = f + O(h), (3) where v 4 and v 5 are the direct neighbors of v in the x and ydirections. 4.. Discretization Near the Interface In the case where one or more of the neighboring nodes involved in the spatial discretization, e.g., v v 3, v 5,v 6 in Fig. 3 or v v 3, v 6 v in Fig. 4, are outside the irregular domain (i.e., φ is positive), the interface intersects between the node v and the neighboring node(s). By enforcing a band of uniform cells along the interface, we guarantee that only uniform cells are cut by the interface, and in this case v is a regular node. Let s consider the D case, as depicted in Fig. 5, in which the interface intersects between v and v 4 at v I. Denote by s I the distance between v I and v. s I is approximated by finding the zero crossing of the quadratic Fig. 5. v is a regular node and the interface Γ intersects between v and v 4 at v I.

13 A Supra-Convergent Finite Difference Scheme 3 interpolant φ = φ + φ x s + φ xxs, where φ is the signed distance level set function. Since v and v are inside the irregular domain with negative values in φ and v 4 is outside with a positive φ, the quadratic interpolant in φ is convex with a positive φ xx, and s I is calculated as: φ x + φx φ xxφ, if φ xx >ɛ, s I = φ xx φ (4) if φ xx ɛ, φ x where ɛ is a small positive number to avoid division by zero. φ x and φ xx are approximated at v using central difference schemes: φ x = s φ 4 φ φ s 4 + s φ 4 s (5) s + s 4 and ( φ4 φ φ xx = φ ) φ. (6) s 4 s s + s 4 These schemes (5) and (6) are second-order accurate because of the locally uniform grid spacing near the interface. Once we know the location of the interface point, the Dirichlet boundary value u I and ρ I are either calculated with an analytic expression or by quadratic interpolation using neighboring nodal values. Then the standard central difference scheme is used to approximate (ρu x ) x : ( ui u (ρu x ) x = ρi + ρ u ) u ρ + ρ s I s s I + s. (7) When the interface intersects between the node and its neighboring nodes in any of the other directions, the discretization can be performed independently in a similar manner, which makes the method straightforward to implement in a dimension-by-dimension framework. In addition, the quadratic interpolation near the interface prevents that the largest error be systematically located near the interface, as pointed out in [4, 3] and guarantees second-order accuracy for both the solution and its gradients. The resulting non-symmetric linear systems are diagonally dominant Heat Equation We consider only the constant coefficient heat equation: T t = k T, (8)

14 3 Chen, Min, and Gibou noting that extension to the variable coefficient heat equation is straightforward. Explicit schemes impose a time step restriction of t (θ x) k for stability, where x is the size of the smallest cell and θ is the smallest cell fraction for cells cut by the interface. Since θ can be arbitrarily small, so is the time step restriction t, which leads to unpractical schemes. Implicit schemes avoid such time step restrictions. We use the Crank Nicolson scheme to discretize the heat Eq. (8): (I ) tk T n+ = (I + ) tk T n, (9) where I is the identity matrix and T n is the solution at the nth time step. The spatial discretization of the Laplace operator in Secs. 4. and 4., is used to approximate T. Since the Crank Nicolson scheme (9) is unconditionally stable and second-order accurate we choose a time step t proportional to x Computing Second-Order Accurate Gradients To compute the gradients to second-order accuracy when T-junctions are present, the ghost values u 4 in Fig. 3 and u 4, u 5 in Fig. 4 are linearly interpolated as in Eqs. (6), (4), and (5). The spurious error caused by the interpolation is successfully removed and the gradients are computed in D (following the notations in Fig. 3) as: u x = u 4 u s 4 u y = u 3 u s 3 s s + s 4 + u u s s s + s 3 + u u s s 4 s + s 4 s 5s 6 s s 4 (s + s 4 ) u yy, s 3 s + s 3, where ( u3 u u yy = + u ) u. s 3 s s + s 3 Following the notations in Fig. 4, the gradients in 3D are calculated as: s 4 u x = u u + u u 4 + s 7s 8 u zz, s s + s 4 s 4 s + s 4 s 4 s + s 4 u y = u u s 5 + u u 5 s + s 9s s u xx s s + s 5 s 5 s + s 5 s 5 s + s 5 + s s s 5 u z = u 6 u s 6 s u zz, s + s 5 s 3 + u u 3 s 3 + s 6 s 3 s s (3) s 3 s 3 + s 6, (3)

15 A Supra-Convergent Finite Difference Scheme 33 where u xx, u yy, and u zz are given by solving Eqs. (8) () simultaneously. In the case where v is a regular node, Eqs. (3) and (3) reduce to the usual weighted average forward and backward differences: in D, and u x = u 4 u s 4 u y = u 3 u s 3 s s 4 + u u, s + s 4 s s + s 4 s + u u s + s 3 s s 4 s 3 s + s 3 (3) u x = u u + u u 4, s s + s 4 s 4 s + s 4 u y = u u s 5 + u u 5 s, s s + s 5 s 5 s + s 5 u z = u 6 u s 3 + u u 3 s 6 (33) s 6 s 3 + s 6 s 3 s 3 + s 6 in 3D. For nodes next to the interface, interface nodes are used in Eqs. (3) and (33) instead of those neighboring nodes that are outside the domain. We note that the calculation of the gradient involves the same cells as those used in the discretization of the Poisson or heat equation, hence preserving the locality and ease of implementation of the method. s 5. EXAMPLES In this section, we present numerical evidence that confirms the schemes described in this paper produce second-order accuracy in the L and L norms for both the solution and its gradients. We use meshes for which the difference of level between adjacent cells can be greater than one, illustrating the fact that the method is supra-convergent on nongraded adaptive grids. The linear systems of equations are solved using the stabilized bi-conjugate gradient (Bi-CGSTAB) method with an incomplete LU preconditioner (see, e.g., [8]). In these examples, we solve the poisson or heat equation on Ω only, noting that the procedure to obtain the solution on Ω+ is similar. We denote by min resoln and max resoln the minimum and maximum grid resolution corresponding to the coarsest and finest cells in the grid, respectively. Specifically, if the cell size is denoted by x, then x = L/min resoln for the coarsest cell and x = L/max resoln for the finest cell, where L is the length of the computational domain. In the following examples, we start with an initial adaptive grid, e.g., with (min resoln, max resoln)=(6, 8), and then recursively split each cell into four (in D) or eight cells (in 3D), which effectively

16 34 Chen, Min, and Gibou increases min resoln and max resoln by one, to obtain the convergence test results for the solution and its gradients. Other starting values of (min resoln, max resoln) can also be used to obtain similar results. We also assume the variable coefficients and the Dirichlet boundary values are defined on grid nodes. Therefore, the variable coefficients and the Dirichlet boundary values on the interface points are calculated by quadratic interpolation using neighboring nodal values. 5.. Poisson Equation 5... Example Consider (ρ u) = f on Ω = [, ] [, ] with an exact solution of u = e (x +y.75 ), where ρ =. The interface is a circle on which φ = x + y.75=. In this example, large solution gradients exist near the circular interface where we impose the finest grid resolution, while the solution near the center is smoother, where we can impose coarser grid resolution. We generate the adaptive grids by proximity to the interface, as described by the criterion in (4). Figure 6 depicts the adaptive Cartesian grid with (min resoln, max resoln) = (6, 64), as well as the interface. The numerical solution on this grid is plotted in Fig. 7. Numerical Fig. 6. Domain Ω = [, ] [, ] and the spatial discretization in example 5.. with (min resoln, max resoln) = (6, 64).

17 A Supra-Convergent Finite Difference Scheme 35 u y.5.5 x Fig. 7. Graph of the solution in example 5.., where (min resoln, max resoln) = (6, 64). The dots represent the approximate solution and the mesh represents the exact solution. Table I. Accuracy Results for the Solution u in Example 5.. (min resoln, max resoln) L error Order L error Order (3, 8) (64, 56) (8, 5) (56, 4) accuracy test results for the solution and its gradients are given in Tables I and II, respectively. In many applications (e.g., Stefan type of problems), high-order accuracy of the solution s gradients at the interface is desired. In this example, numerical accuracy test is also performed for the gradient at Table II. Accuracy Results for the Solution s Gradients u in Example 5.. (min resoln, max resoln) L error Order L error Order (6, 8) (3, 56) (64, 5) (8, 4)

18 36 Chen, Min, and Gibou Table III. Accuracy Results for the Solution s Gradients u on the Interface in Example 5.. (min resoln, max resoln) L error Order L error Order (6, 64) (3, 8) (64, 56) (8, 5) the interface, which is obtained by second-order extrapolation of the gradient from Ω to Ω + followed by quadratic interpolation on the interface. Table III demonstrates that second-order accuracy of the solution s gradients can be obtained at the interface in both the L and L norms. The resulting linear system is non-symmetric but diagonally dominant, which we solve with the stabilized bi-conjugate gradient (Bi-CGSTAB) method with an incomplete LU preconditioner. In [], Gibou et al. proposed a symmetric solver for the Poisson and heat equation on irregular domains and uniform grids. The advantage of this method is the symmetry of the linear system, which can be inverted with a number of fast methods, such as the conjugate gradient method with an incomplete Cholesky preconditioner [8]. However, the solutions gradients are only first-order accurate. We present a brief comparison between the method in Gibou et al. [] with that of the present paper. We find that the present method compares favorably. We plot the maximum solution and gradient errors as a function of the number of nodes inside the irregular domain in Figs. 8 and 9, respectively. It can be seen that while both methods yield second-order accurate solutions, the number of nodes in our method is about one order of magnitude less than that in [], if the same error is to be obtained. Our method is also advantageous in that the maximum gradient error, which occurs near the interface in this example, decays quadratically, while it decays linearly in []. This is consistent with the argument in [4] about the method in []. Since different solvers of the linear system are used in these two methods, we also plot the maximum solution and gradient errors as a function of the computational time in Figs. and, respectively. Our method is obviously more efficient, especially when high accuracy of the gradient is desired. In these plots, the adaptive grid resolutions are (min resoln, max resoln) = (6, 64), (3, 8), (64, 56), (8, 5), and the uniform grid resolutions are 64 64, 8 8, 56 56, 5 5.

19 A Supra-Convergent Finite Difference Scheme 37 maximum solution error on adaptive grids maximum solution error on uniform grids error number of nodes in the irregular domain Fig. 8. Log log plot of the solution error as a function of the number of nodes in the irregular domain for example 5... Our method on adaptive grids is compared with that on uniform grids in []. We also note that in term of speed, one should invest on the implementation of the fast linear solver using for example multigrid methods, which are significantly more efficient than Bi-CGSTAB Example Consider (ρ u) = f on Ω = [, ] [, ] with an exact solution of u = sin(x) sin(y) + e (x +y ), where ρ =. The interface is a circle on which φ = x + y.75 =. Large solution gradients are expected near the center of the circle where we put finer resolution in our adaptive grids, while solutions near the interface is smoother and we put coarser resolution there. An adaptive Cartesian grid with (min resoln, max resoln) = (6, 8), as well as the interface, are illustrated in Fig.. The numerical solution on this grid is plotted in Fig. 3. Numerical accuracy test results for the solution and its gradients are given in Tables IV and V, respectively.

20 38 Chen, Min, and Gibou error 3 maximum gradient error on adaptive grids maximum gradient error on uniform grids number of nodes in the irregular domain Fig. 9. Log log plot of the gradient error as a function of the number of nodes in the irregular domain for example 5... Our method on adaptive grids is compared with that on uniform grids in []. Table IV. Accuracy results for the solution u in example 5.. (min resoln, max resoln) L error Order L error Order (6, 8) (3, 56) (64, 5) (8, 4) Table V. Accuracy Results for the Solution s Gradients u in Example 5.. (min resoln, max resoln) L error Order L error Order (6, 8) (3, 56) (64, 5) (8, 4)

21 A Supra-Convergent Finite Difference Scheme 39 maximum solution error on adaptive grids maximum solution error on uniform grids error 3 4 time (seconds) Fig.. Log log plot of the solution error as a function of the computational time for example 5... Our method on adaptive grids is compared with that on uniform grids in [] Example 3 Consider (ρ u) = f on Ω = [, ] [, ] with an exact solution of u = sin(πx) sin(πy), where ρ = sin(xy) +. This equation is solved within an annulus between two circles with radii being.5 and.75. In this example, and all the following examples in this paper, we generate the adaptive grids by the criterion in 4, and put the finest resolution in a uniform band along the interface. A non-graded Cartesian grid with (min resoln, max resoln) = (8, 8), as well as the interface, are illustrated in Fig. 4. The numerical solution on this grid is plotted in Fig. 5. Numerical accuracy test results for the solution and its gradients are given in Tables VI and VII, respectively Example 4 Consider (ρ u) = f on Ω = [, ] [, ] with an exact solution of u = e xy, where ρ = x + y. The interface is diamond shaped, given by the set of points where φ = y +.5 x.75 =. A non-graded Cartesian grid with (min resoln, max resoln) = (8, 8), as well as the interface, are

22 4 Chen, Min, and Gibou error 3 maximum gradient error on adaptive grids maximum gradient error on uniform grids time (seconds) Fig.. Log log plot of the gradient error as a function of the computational time for example 5... Our method on adaptive grids is compared with that on uniform grids in []. Fig.. Domain Ω = [, ] [, ] and the spatial discretization in example 5.. with (min resoln, max resoln) = (6, 8).

23 A Supra-Convergent Finite Difference Scheme 4.5 u.5 y.5.5 x.5 Fig. 3. Graph of the solution in example 5.., where (min resoln, max resoln) = (6, 8). The dots represent the approximate solution and the mesh represents the exact solution. Fig. 4. Domain Ω = [, ] [, ] and the spatial discretization in example 5..3 with (min resoln, max resoln) = (8, 8).

24 4 Chen, Min, and Gibou.5 u.5.5 y x Fig. 5. Graph of the solution in example 5..3, where (min resoln, max resoln) = (8, 8). The dots represent the approximate solution and the mesh represents the exact solution. Table VI. Accuracy Results for the Solution u in Example 5..3 (min resoln, max resoln) L error Order L error Order (8, 8) (6, 56) (3, 5) (64, 4) Table VII. Accuracy Results for the Solution s Gradients u in Example 5..3 (min resoln, max resoln) L error Order L error Order (8, 8) (6, 56) (3, 5) (64, 4) illustrated in Fig. 6. The numerical solution on this grid is plotted in Fig. 7. Numerical accuracy test results for the solution and its gradients are given in Tables VIII and IX, respectively.

25 A Supra-Convergent Finite Difference Scheme 43 Fig. 6. Domain Ω = [, ] [, ] and the spatial discretization in examples 5..4 and 5.. with (min resoln, max resoln) = (8, 8). u y.5.5 x.5 Fig. 7. Graph of the solution in example 5..4, where (min resoln, max resoln) = (8, 8). The dots represent the approximate solution and the mesh represents the exact solution Example 5 Consider (ρ u) = f on Ω = [, ] [, ] with an exact solution of u = cos(πx) sin(πy), where ρ = e xy. The interface is a cardioid given by the set of points where φ = ((3(x + y ) x) x y )/47 =. Note that

26 44 Chen, Min, and Gibou Table VIII. Accuracy Results for the Solution u in Example 5..4 (min resoln, max resoln) L error Order L error Order (8, 8) (4, 8) (5, 9) (6, ) Table IX. Accuracy Results for the Solution s Gradients u in Example 5..4 (min resoln, max resoln) L error Order L error Order (8, 8) (4, 8) (5, 9) (6, ) Table X. Accuracy Results for the Solution u in Example 5..5 (min resoln, max resoln) L error Order L error Order (8, 8) (4, 8) (5, 9) (6, ) the interface is not even Lipschitz continuous and the singular point of the interface is the cusp point (, ). A non-graded Cartesian grid with (min resoln, max resoln) = (8, 8), as well as the interface, are illustrated in Fig. 8. The numerical solution on this grid is plotted in Fig. 9. Numerical accuracy test results for the solution and its gradients are given in Tables X and XI, respectively Example 6 Consider (ρ u) = f on Ω = [, ] [, ] with an exact solution of u=e xy, where ρ =x +y. The interface is star shaped, given by the set of points where φ = r.5 y5 +5x 4 y x y 3 =, and r = x 3r + y. A nongraded Cartesian grid with (min resoln, 5 max resoln) = (8, 8), as well as

27 A Supra-Convergent Finite Difference Scheme 45 Fig. 8. Domain Ω = [, ] [, ] and the spatial discretization in examples 5..5 and 5..3 with (min resoln, max resoln) = (8, 8)..5 u.5.5 y.5 Fig. 9. Graph of the solution in example 5..5, where (min resoln, max resoln) = (8, 8). The dots represent the approximate solution and the mesh represents the exact solution..5 x.5

28 46 Chen, Min, and Gibou Table XI. Accuracy Results for the Solution s Gradients u in Example 5..5 (min resoln, max resoln) L error Order L error Order (8, 8) (4, 8) (5, 9) (6, ) Fig.. Domain Ω = [, ] [, ] and the spatial discretization in examples 5..6 and 5..4 with (min resoln, max resoln) = (8, 8). the interface, are illustrated in Fig.. The numerical solution on this grid is plotted in Fig.. Numerical accuracy test results for the solution and its gradients are given in Tables XII and XIII, respectively Example 7 Consider (ρ u) = f on Ω = [, ] [, ] with an exact solution of u = cos(πx) sin(πy), where ρ = + sin(πx) cos(πy). The interface is given by the set of points where φ = 6y 4 x 4 3y + 9x =. A nongraded Cartesian grid with (min resoln, max resoln) = (8, 8), as well as

29 A Supra-Convergent Finite Difference Scheme 47 u y.5.5 x.5 Fig.. Graph of the solution in example 5..6, where (min resoln, max resoln) = (8, 8). The dots represent the approximate solution and the mesh represents the exact solution. Table XII. Accuracy Results for the Solution u in Example 5..6 (min resoln, max resoln) L error Order L error Order (8, 8) (4, 8) (5, 9) (6, ) Table XIII. Accuracy Results for the Solution s Gradients u in Example 5..6 (min resoln, max resoln) L error Order L error Order (8, 8) (4, 8) (5, 9) (6, ) the interface, are illustrated in Fig.. The numerical solution on this grid is plotted in Fig. 3. Numerical accuracy test results for the solution and its gradients are given in Tables XIV and XV, respectively.

30 48 Chen, Min, and Gibou Fig.. Domain Ω = [, ] [, ] and the spatial discretization in examples 5..7 and 5..5 with (min resoln, max resoln) = (8, 8). u y x.5 Fig. 3. Graph of the solution in example 5..7, where (min resoln, max resoln) = (8, 8). The dots represent the approximate solution and the mesh represents the exact solution Example 8 Consider a 3D Poisson equation (ρ u) = f on Ω = [.75,.75] [.75,.75] [.75,.75], with an exact solution of u = exp( x y z ), where ρ =+sin(x +y +z). The interface is the surface of an ellipsoid

31 A Supra-Convergent Finite Difference Scheme 49 Table XIV. Accuracy Results for the Solution u in Example 5..7 (min resoln, max resoln) L error Order L error Order (8, 8) (4, 8) (5, 9) (6, ) Table XV. Accuracy Results for the Solution s Gradients u in Example 5..7 (min resoln, max resoln) L error Order L error Order (8, 8) (4, 8) (5, 9) (6, ) Table XVI. Accuracy Results for the Solution u in Example 5..8 (min resoln, max resoln) L error Order L error Order (4, 6) (8, 3) (6, 64) (3, 8) where φ = x + 9/4y + 4z.5 =. A non-graded Cartesian grid with (min resoln, max resoln) = (4, 3), as well as the interface, are illustrated in Fig. 4. The numerical accuracy test results for the solution and its gradients are given in Tables XVI and XVII, respectively. 5.. Heat Equation We consider Eq. () in two and three spatial dimensions, where ρ is a (possibly different) constant on each subdomain. Again, we only compute solutions on Ω. Dirichlet boundary conditions are applied on the interface. The time step in the Crank Nicolson scheme is chosen as t = x/, where x is the size of the smallest cell, so that an overall second-order accuracy is obtained. Since the solutions in the following examples decay

32 5 Chen, Min, and Gibou Fig. 4. Domain Ω = [.75,.75] [.75,.75] [.75,.75] and the spatial discretization in example 5..8 with (min resoln, max resoln) = (4, 3). Table XVII. Accuracy Results for the Solution s Gradients u in Example 5..8 (min resoln, max resoln) L error Order L error Order (4, 6) (8, 3) (6, 64) (3, 8) exponentially, we choose the final time as t =.5, so that the solution at the final time is not too small. Other choices of the final time would yield similar results Example Consider u t = (ρ u) on Ω = [ π,π] [ π,π] with an exact solution of u = e ρt sin(x) cos (y), where ρ =. The interface is given by the set of points where φ = x + (y ).5 =. Dirichlet boundary condition is also applied on Ω. An adaptive Cartesian grid with (min resoln, max resoln) = (8, 64), as well as the interface, are illustrated in

33 A Supra-Convergent Finite Difference Scheme 5 Fig. 5. Domain Ω = [, ] [, ] and the spatial discretization in example 5.. with (min resoln, max resoln) = (8, 64)..5 u.5 y Fig. 6. Graph of the solution in example 5.., where (min resoln, max resoln) = (8, 64). The dots represent the approximate solution and the mesh represents the exact solution. x Fig. 5. The numerical solution at t =.5 on this grid is plotted in Fig. 6. Numerical accuracy test results for the solution and its gradients are given in Tables XVIII and XIX, respectively.

34 5 Chen, Min, and Gibou Table XVIII. Accuracy Results for the Solution u in Example 5.. (min resoln, max resoln) L error Order L error Order (8, 64) (6, 8) (3, 56) (64, 5) Table XIX. Accuracy Results for the Solution s Gradients u in Example 5.. (min resoln, max resoln) L error Order L error Order (8, 64) (6, 8) (3, 56) (64, 5) Example Consider u t = (ρ u) on Ω = [, ] [, ] with an exact solution of u=e π ρt cos(πx) cos(πy), where ρ =. The interface is the same as in example The numerical solution at t =.5 on a grid with (min resoln, max resoln) = (8, 64) is plotted in Fig. 7. Numerical accuracy test results for the solution and its gradients are given in Tables XX and XXI, respectively. Table XX. Accuracy Results for the Solution u in Example 5.. (min resoln, max resoln) L error Order L error Order (8, 64) (6, 8) (3, 56) (64, 5)

35 A Supra-Convergent Finite Difference Scheme 53 u x 3.5 y.5.5 Fig. 7. Graph of the solution in example 5.., where (min resoln, max resoln) = (8, 64). The dots represent the approximate solution and the mesh represents the exact solution. x.5 Table XXI. Accuracy Results for the Solution s Gradients u in Example 5.. (min resoln, max resoln) L error Order L error Order (8, 64) (6, 8) (3, 56) (64, 5) Example 3 Consider u t = (ρ u) on Ω = [, ] [, ] with an exact solution of u = e ρt cos(x) cos(y), where ρ =. The interface is the same as in example The numerical solution at t =.5 on a grid with (min resoln, max resoln) = (8, 64) is plotted in Fig. 8. Numerical accuracy test results for the solution and its gradients are given in Tables XXII and XXIII, respectively Example 4 Consider u t = (ρ u) on Ω = [, ] [, ] with an exact solution of u = e ρt (sin(x) + cos(y)), where ρ = 8. The interface is the same as in example The numerical solution at t =.5 on a grid with

36 54 Chen, Min, and Gibou u y.5 Fig. 8. Graph of the solution in example 5..3, where (min resoln, max resoln) = (8, 64). The dots represent the approximate solution and the mesh represents the exact solution..5 x.5 Table XXII. Accuracy Results for the Solution u in Example 5..3 (min resoln, max resoln) L error Order L error Order (8, 64) (6, 8) (3, 56) (64, 5) Table XXIII. Accuracy Results for the Solution s Gradients u in Example 5..3 (min resoln, max resoln) L error Order L error Order (8, 64) (6, 8) (3, 56) (64, 5) (min resoln, max resoln) = (8, 64) is plotted in Fig. 9. Numerical accuracy test results for the solution and its gradients are given in Tables XXIV XXVI, respectively.

37 A Supra-Convergent Finite Difference Scheme u..5.5 y.5 Fig. 9. Graph of the solution in example 5..4, where (min resoln, max resoln) = (8, 64). The dots represent the approximate solution and the mesh represents the exact solution..5 x.5 Table XXIV. Accuracy Results for the Solution u in Example 5..4 (min resoln, max resoln) L error Order L error Order (8, 64) (6, 8) (3, 56) (64, 5) Table XXV. Accuracy Results for the Solution s Gradients u in Example 5..4 (min resoln, max resoln) L error Order L error Order (8, 64) (6, 8) (3, 56) (64, 5) Example 5 Consider u t = (ρ u) on Ω = [, ] [, ] with an exact solution of u = e π ρt cos(πx) sin(πy), where ρ =.. The interface is the same as in example The numerical solution at t =.5 on a grid

38 56 Chen, Min, and Gibou u.. y x.5 Fig. 3. Graph of the solution in example 5..5, where (min resoln, max resoln) = (8, 64). The dots represent the approximate solution and the mesh represents the exact solution. Table XXVI. Accuracy Results for the Solution s Gradients u in Example 5..4 (min resoln, max resoln) L error Order L error Order (8, 64) (6, 8) (3, 56) (64, 5) Table XXVII. Accuracy Results for the Solution u in Example 5..5 (min resoln, max resoln) L error Order L error Order (8, 64) (6, 8) (3, 56) (64, 5) with (min resoln, max resoln) = (8, 64) is plotted in Fig. 3. Numerical accuracy test results for the solution and its gradients are given in Tables XXVII and XXVIII, respectively.

39 A Supra-Convergent Finite Difference Scheme 57 Table XXVIII. Accuracy Results for the Solution s Gradients u in Example 5..5 (min resoln, max resoln) L error Order L error Order (8, 64) (6, 8) (3, 56) (64, 5) Fig. 3. Domain Ω = [.75,.75] [.75,.75] [.75,.75] and the spatial discretization in example 5..6 with (min resoln, max resoln) = (4, 3) Example 6 Consider a 3D heat equation u t = (ρ u) on Ω = [.75,.75] [.75,.75] [.75,.75] with an exact solution of u = e t (sin(x) + cos(y) + cos(z). The interface is the surface of a sphere on which φ = x + y + z.7. A non-graded Cartesian grid with (min resoln, max resoln) = (4, 3), as well as the interface, are illustrated in Fig. 3. Numerical accuracy test results for the solution and its gradients are given in Tables XXIX and XXX, respectively.

40 58 Chen, Min, and Gibou Table XXIX. Accuracy Results for the Solution u in Example 5..6 (min resoln, max resoln) L error Order L error Order (4, 6) (8, 3) (6, 64) (3, 8) Table XXX. Accuracy Results for the Solution s Gradients u in Example 5..6 (min resoln, max resoln) L error Order L error Order (4, 6) (8, 3) (6, 64) (3, 8) CONCLUSIONS We have proposed finite difference algorithms for the variable coefficient Poisson equation as well as for the heat equation on irregular domains and non-graded Cartesian grids. These schemes leverage both adaptivity strategy and the quadratic treatment of the Dirichlet boundary condition near the interface and produce second-order accuracy for the solution and its gradients. T-junction nodes neighboring the irregular interface are excluded by imposing that cells on the interface have the finest resolution. Sampling the solution at the nodes produces efficient and simple procedures that can be applied in a dimension-by-dimension framework. At T-junctions, linear interpolations are used to generate intermediate ghost values used in the discretizations. These intermediate values introduce spurious O() errors that are successfully removed by linear combinations of the discretizations in the transversal directions. For nodes neighboring the interface, a quadratic fitting of the signed distance level set function is used to find the location of the interface, and the Dirichlet boundary value is obtained either explicitly when a analytic formula is known or by quadratic interpolation of the neighboring nodal values. The calculation of the solution s gradients involves the same cells as those used in the discretization of the Poisson or heat equation, hence preserving the locality and ease of implementation of the method. We have presented