Long time stability of regularized PML wave equations

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1 Long time stability of regularized PML wave equations Dojin Kim Yonghyeon Jeon Philsu Kim Abstrat In this paper, we onsider two dimensional aousti wave equations in an unbounded domain and introdue a modified model of the lassial perfetly mathed layer (PML. In the lassial PML model, an unexpeted and exponential inrease in energy is observed in the long-time simulation after the solution reahes a quiesent state. To address suh an instability, we provide a regularization tehnique to a lower order regularity term employed in the auxiliary variable in the lassial PML model. The well-posedness of the regularized system is analyzed with the standard Galerkin method based on the energy analysis, and the numerial stability of staggered finite differene method for its disretization is provided by using von Neumann stability analysis. To support the theoretial results, under various thikness and damping values, we demonstrate a long-time stability of aousti waves in the omputational domain. I. INTRODUCTION It is quite important to effetively trunate an unbounded domain in wave propagation simulations in free spae, where the perfetly mathed layer (PML methods that surround the domain of interest with thin artifiial absorbing layers are popularly used in easy and effetive ways. After the method was introdued by J. P. Bérenger [6], whih involves splitting a field into two nonphysial eletromagneti fields, several studies were onduted regarding the PML method and its modified reformulations in many different wave-type equations. These inlude Maxwell s equations [9], [6], elastodynamis [7], [], linearized Euler equations [], [4], [], [3], Helmholtz equations [8], and other types of wave equations [], [6], [8]. Most PML models by the splitting tehnique, named a split PML method, yield a hyperboli system of first order partial differential equations [8], [], [3], [6], [9]. It is known that the split PML models demonstrate exellent overall performanes from the viewpoint of appliations. However, it was pointed out in [3], [5], [4] that Bérenger s split, as well as other split models, transform Maxwell s equations from being strongly hyperboli into weakly hyperboli. These transforms imply a transition from strong to weak well-posedness in the Cauhy problem and may lead to ill-posedness under low-order damping funtions or thin layers [5]. The authors of [4], [] mention that the use of artifiial dissipation is neessary to stabilize the numerial sheme of suh formulations for long-time simulations. The resulting onerns about the well-posedness and stability of the split PML models have prompted the development of other PMLs. Some examples of suh developments, without splitting the fields, inlude un-split PML models using onvolution integrals [4], [5] and auxiliary variables [], [3], []. In ontrast to the split PML models, it is known that the un-split PML wave equations are more effetive at time disretization [] and does not make the use of additional memory for the nonphysial field variables. However, it has been found that the un-split PML models are suseptible to developing gradual instabilities in long-time simulations [4], [8]. These issues are the motivation for the mathematial study of the well-posedness and stability for the un-split PML models. Contrary to the many existing laims, the PML model may generate instability when the solution is quiesent or nearly quiesent in the PML layers, whih is demonstrated numerially in Setion II. This instability auses the wave energy to exponentially inrease in the omputational domain, whih depends on the damping and layer thikness in the experiments. The main ontribution of this study is not only to introdue a regularized system of the seond order PML aousti wave equation that exhibits well-posedness without losing the nonrefletion property of PMLs, but also to demonstrate its numerial stability. To onstrut the system, we adopt a regularization tehnique in the term q that has a lower regularity, to regularize the PML model for the Maxwell equation, where q is the auxiliary variable (see (II.3. The standard Galerkin approximation and energy estimation of the solution are used to show the well-posedness of the regularized system. For its numerial sheme, we use a family of finite differene shemes using half-step staggered grids in spae and time with entral finite differenes that maintain the seond order approximation in both spae and time, respetively. A onrete von Neumann stability analysis for the numerial sheme indiates that the sheme is stable under the Courant-Friedrihs-Lewy (CFL ondition. The novel features of this study inlude the good performane of the solution that presents a long-time stability ompared to the lassial PML model, whih is numerially illustrated by presenting the energy behavior for the solutions in Setion IV. ISSN:

2 II. MOTIVATION OF STUDY The target problem we onsider is a general seond order aousti wave equation with a variable sound speed (x > desribed by u tt (x, t (x u(x, t =, (x, t R (, T ] (II. with initial onditions u(, = u and u t (, =, where supp(u is a subset of a domain Ω inluded in the omputational domain Ω omp := [ a, a] [ b, b] in R for some a, b >. Here, T > and the sound speed (x is assumed to be bounded by where x k = x, y (k =,, β =,,, σ is a given onstant, and L denotes the thikness of the layers. The smooth initial value u for our numerial tests is adopted as { u (x, y = e.6((x x +(y y if (x, y Ω omp, otherwise, (II.6 and a onstant sound speed (x, y = is assumed. With = 7 = 6 = 5 = 4 < (x <. (II. = 3 Suppose that a domain Ω = [ a L x, a+l x ] [ b L y, b+ L y ] onsists of Ω omp surrounded by PMLs, where L x, L y >. Using a omplex oordinate streth, Grote and Sim [] introdued the following two dimensional aousti PML wave model: find (u, q satisfying, for (x, t Ω (, T ], { u tt + αu t + βu q u =, (II.3 q t + A q + B u = with initial and boundary onditions u(, := u, u t (, := u =, q(, := q =, u(x, Ω =, where q denotes the auxiliary variable and the oeffiients ( are given by ( α := σx+σy, β := σxσy, σx σx σ A =, and B = y σ y σ y σ x. Here, the damping terms σ x := σ x (x and σ y := σ y (y are assumed to be nonnegative funtions that vanish in the omputational domain Ω omp in the sense of the analytial ontinuation of the PML [7]. It is noted that the Cauhy problem for the model problem (II.3 is strongly stable in []. We also note that the model problem turns into an initial boundary problem by the artifiial arving of the omputational domain, whih generates the boundary. From this artifiial building of the boundary, therefore, the wave propagation should be affeted by its ondition. To demonstrate these effets, we use the energy method, introdued in [8], and numerially examine the well-posedness and stability of the model (II.3 by observing the behavior of the aousti wave energy defined by E(t = Ω omp ( u t(t + u(t u(t dx. (II.4 For this purpose, we use the omputational domain Ω omp = [, ] [, ] surrounded by a PML layer with thikness L x = L y = L. In the absorbing layer, we onsider the onstant ase (β = in the following damping funtion of the form;, x k <, ( σ xk (x k = β xk (II.5 σ, x k + L, L Fig. : E(t when a fixed L = and various dampings these onditions, we first disretize equations (II.3 with the staggered finite differene sheme in time and spae under a uniform spatial grid size h = x = y =.5. In addition, we hoose a time step size t of h/3, whih satisfies the CFL ondition (III.5 to guarantee the stability of the staggered finite differene sheme for equation (II.3. The first order bakward and seond order entral finite differenes in time and spae, respetively, are used to disretize the energy E(t n of (II.4 at eah time step t n L= L=.75 L- L= Fig. : E(t when σ = 5, β = and various thiknesses L We investigate the behavior of the energy for a longtime simulation at time t n = aording to the thikness of the layers and magnitude of the damping. The numerial results are displayed in Fig. and : the energy for a fixed thikness L = with various dampings σ = 3, 4, 5, 5, 6, 7 (Fig. and the energy for a fixed damping σ = 5 with various thiknesses L =,,.75, (Fig.. The results indiate that the numerial stability of the PML model is quite suseptible to both the thikness of the layers and magnitude of the damping, as noted in [7]. Furthermore, in the long-time simulation, one an observe that the wave energy unexpetedly and exponentially inrease in the omputational domain. These phenomena indiretly show that the stability argument of the energy developed by [8] is not learly answered. Furthermore, the experiments indiate that the stability analysis [] for the Cauhy problem for the model (II.3 is not suffiient. The unexpeted phenomena detailed above is likely to our due to the non-physial auxiliary variable q, or a lower regularity term q in the layers of the un-split PML model (II.3. This experiment makes it neessity to further develop ISSN:

3 the PML model, whih is a motivation to the study. III. REGULARIZED SYSTEM Based on the motivation disussed in Setion II, the aim of this setion is to introdue a regularized system overoming the instability that ours in the lassial PML model (II.3 for the aousti wave equation (II.. Let H (Ω = {ϕ : ϕ, x ϕ, y ϕ L (Ω} and H (Ω denote the Sobolev spae and dual spae of H (Ω, respetively. First, note that q H (Ω, whih does not have a suffiient regularity so that the weak solution (u, q of (II.3 is inluded in H (Ω L (Ω, where L (Ω := [L (Ω]. To handle this problem of lower regularity, we define a linear bounded regularization operator δ ε satisfying δ ε as ε and δ ε (ϕ L (Ω C δε ϕ H (Ω for some C δε >. Following [9], [], we introdue a regularized system of the lassial PML model (II.3 by using δ ε in the term q, whih is given by, for (x, t Ω (, T ] u tt + αu t + βu δ ε q u =, (III. q t + A q + B u = with initial and boundary onditions u(, := u, u t (, := u =, q(, := q =, u(x, Ω =. The remainder details the analysis of the well-posedness of the solution to the regularized system (III. based on the energy estimation under the assumption of the dampings σ x, σ y L (Ω. A. Well-posedness of weak solution We assume that the damping funtions satisfy σ x, σ y L (Ω and (x = in the layers of the PML model (II.3. Under these assumptions, we define the weak solution of (III. in the sense that u L (, T ; H (Ω, q L (, T ; L (Ω with u t L (, T ; L (Ω, u tt L (, T ; H (Ω, q t L (, T ; L (Ω, whih satisfies { u tt, w + (αu t + βu δ ε q, w + ( u, w =, ( q t, v + (A q, v + (B u, v = (III. for eah w H (Ω, v L (Ω, and almost everywhere t T and the initial data satisfy (u(, w = (u, w, u t (, w = (u, w, ( q(, v = ( q, v for eah w H (Ω, v L (Ω. Here,, denotes the duality pairing between H (Ω and H (Ω, and (, is the inner produt in L (Ω. In addition, the time derivatives are understood in a distributional sense. To investigate the weak solution of (III. that satisfies (III. with the initial ondition, we an show the well-posedness of the regularized system (III. using the standard Galerkin approximation and estimate the energy of the solution. Let U k be the subspae generated by the - weighted orthogonal basis of H (Ω in the sense that ( w j, w k + ( wj, w k = if j k. Let us also denote Q k, whih is the spae generated by the smooth funtions { v, v,, v k } suh that { v k : k N} is an orthonormal basis of L (Ω. We onstrut approximate solutions ( u k, q k, k =,, 3,, in the form u k (t = k gj k (tw j, q k (t = j= k h k j (t v j, j= (III.3 whose oeffiients gj k(t, hk j (t, j =,,, k, are hosen so that gj k( = (u, w j, (gj k t( = (u, w j, h k j ( = ( q, v j and {( ( u k tt + αu k t + βu k δ ε q k, w j + u k, w j =, ( ( ( q k t, v j + A q k, v j + B u k, v j = (III.4 are satisfied for all w j U k, v j Q k, j =,, k. For eah integer k =,,, the standard theory of ordinary differential ( equations guarantees the existene of the approximation u k (t, q k (t satisfying (III.3 and (III.4. The following theorem gives a uniform bound of energy of the approximate solutions (III.3, whih allows us to send k. Theorem 3.: There exists a onstant C T > that depends only on σ x, σ y, Ω, and T suh that for k max E k(t + u k L + tt t T (,T ;H (Ω q k L t (,T ;L (Ω C T ( u H (Ω + u L (Ω + q L (Ω, where the energy E k (t is defined by E k (t = uk t (t L (Ω + uk (t L (Ω + q k (t L (Ω. Passing to the limit from the bounds, we obtain the following theorem. Theorem 3.: (Existene and Uniqueness Assume that the initial onditions (u, u, q H (Ω L (Ω L (Ω. Then, the system (III.4 has a unique weak solution provided by σ x, σ y L (Ω. (see [9] for detail proof of uniqueness. B. Numerial Sheme For the staggered finite differene method, we use a family of finite differene shemes [] with half-step staggered grids in spae and time. All spatial derivatives are disretized with the entered finite differenes over two or three ells, whih guarantees a seond order approximation in spae. For the time disretization, we also use the entered finite differenes for the first and seond order time derivatives on a uniform mesh, whih is also of the seond order approximation in time. Based on the standard von Neumann stability analysis tehnique, we analyze the stability of the numerial sheme and obtain its CFL ondition. To obtain the stability ondition of the staggered finite differene sheme defined above, we restrit our onern to the onstant damping ase with σ x = σ y = σ for simpliity in our analysis. The stability ondition for the sheme in the omputational domain is as follows. Remark 3.3: The CFL ondition of the staggered sheme in the omputational area (i.e., σ x = σ y = is t h ISSN:

4 for x = y = h from the standard von Neumann stability analysis tehnique. Generally the stability ondition for the staggered finite differene sheme an be obtained as follows. Theorem 3.4: Assume that σ x = σ y = σ > and the sound speed are onstants. Then, the disrete sheme is stable if the CFL ondition t h ( + σ h 8 / (III.5 is satisfied for x = y = h. (see [9] for detail proof IV. NUMERICAL RESULT The aim of this setion is to provide numerial evidene of the well posedness of the regularized system and disuss the numerial stability in the long-time simulation for the staggered finite differene method. To do this, we demonstrate the behavior of the aousti wave energy E(t defined in (II.4 in the omputational domain. For the numerial simulation, we use the same initial ondition defined by (II.6 and, in the absorbing layer, the damping funtion of the form in (II.5. For the sound speed (x, y, we onsider the onstant form as well as variable sound speeds. For a omparison with the numerial results for the lassial PML system (II., we first simulate the regularized system under the same onditions for the damping σ with β = and thikness L of the layers shown in Setion II. The numerial results of formulae (II. and (III. are displayed in Fig. 3 and 4. As shown in the figures, it an be observed that the wave energy E(t for the regularized system exhibits an outstanding stability performane in the long-time simulation independent of both the magnitude of the damping σ and thikness L. For further investigation, using variable sound speeds, the nononstant damping values of (II.4 with β =,, and the different magnitudes σ = 3, 5, 8, we examine the long time behaviors of E(t for formulae (II. and (III., and display the numerial results in Fig. 5. It an be observed that an unexpeted and exponential growth of E(t of the lassial PML model ours in the long time simulation, similar to the ase of β = shown in Setion II. In ontrast to the lassial one, however, the wave energy E(t of the regularized system are onsistently stable in the long-time simulation regardless of the damping, thikness, and sound speed. In summary, the regularized system demonstrates a good numerial stability performane in terms of the measure E(t for a long-time simulation under the various damping, layer thikness, and sound speed values. This provides proof of the well-posedness of the developed system and numerial stability for the finite differene method. These numerial results show the extent to whih the solution of the lassial PML model is affeted by the regularization δ ɛ q for the divergene of the auxiliary variable q, whih has a lower regularity in H (Ω. Contrary to the theoretial laim of [8], however, the lassial PML model presents an instability of E(t, for ertain damping and thikness values, whih may ome from the lower regularity of q. 3.5 Regularized system L=,,.75, Classial PML Model : L= Classial PML model: L= Classial PML model: L=.75 Classial PML model: L= Fig. 3: E(t when σ = 5, β = and various thikness L Classial PML model: = 7 Classial PML model: = 6 Classial PML Classial PML model: = 4 model: = 5 Regularized system = 3, 4, 5, 6, Fig. 4: E(t when L =, β = and various damping σ Classial PML model : =, =, L= Regularized system: =7, =, L= =, =, L= Classial PML model: =7, =, L= Fig. 5: E(t when various damping values and variable sound speed CONCLUSIONS We have introdued a new and effiient formulation related to the aousti wave equation based on the regularization of the un-split PML wave equation. By regularizing the lower order regularity term in the original equation and the standard von Neumann stability analysis, we have ahieved well-posedness as well as numerial stability, even in the long-time simulation, of the solution in the new formulation. We have demonstrated the extent to whih the regularization is important in the long time stability by several numerial tests. We summarize the main novelty and results of this study as follows: ( We have proved the analytial well-posedness of our formulation without any restrition of damping terms; ( several numerial tests suggest that the formulation exhibhits a longtime stability regardless of damping terms, layer thikness, and sound speeds; (3 we have demonstrated that the lower order regularity term in the up-split PML formulation highly affets the long-time stability; this is a strong motivation for the regularization. ACKNOWLEDGEMENTS This researh was supported by the basi siene researh program through the National Researh Foundation of Korea (NRF, and funded by the Ministry of Eduation, Siene and Tehnology (grant number 6RAB36. 4 ISSN:

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