Local Discontinuous Galerkin Methods for the Khokhlov Zabolotskaya Kuznetzov Equation

Transcription

1 J Sci Comut (7 73: DOI.7/s z Local Discontinuous Galerkin Methods for the Khokhlov Zabolotskaya Kuznetzov Equation Ching-Shan Chou Weizhou Sun Yulong Xing He Yang Received: 5 Setember 6 / Revised: May 7 / Acceted: July 7 / Published online: 3 August 7 Sringer Science+Business Media, LLC 7 Abstract Khokhlov Zabolotskaya Kuznetzov (KZK equation is a model that describes the roagation of the ultrasound beams in the thermoviscous fluid. It contains a nonlocal diffraction term, an absortion term and a nonlinear term. Accurate numerical methods to simulate the KZK equation are imortant to its broad alications in medical ultrasound simulations. In this aer, we roose a local discontinuous Galerkin method to solve the KZK equation. We rove the L stability of our scheme and conduct a series of numerical exeriments including the focused circular short tone burst excitation and the roagation of unfocused sound beams, which show that our scheme leads to accurate solutions and erforms better than the benchmark solutions in the literature. Keywords Local discontinuous Galerkin method KZK equation Stability analysis In honor of Professor Shu s 6th birthday. CSC is artially suorted by NSF Grant DMS YX is artially suorted by NSF Grant DMS-6 and ONR Grant N B Ching-Shan Chou chou@math.ohio-state.edu Weizhou Sun sun.435@osu.edu Yulong Xing xingy@ucr.edu He Yang yang.67@osu.edu Deartment of Mathematics, The Ohio State University, Columbus, OH 43, USA Deartment of Mathematics, University of California Riverside, Riverside, CA 95, USA

2 594 J Sci Comut (7 73: Introduction Khokhlov Zabolotskaya Kuznetzov (KZK equation is one of the most widely used models in medical ultrasound simulations. KZK equation can be regarded as a arabolic aroximation of the more general Westervelt equation, and it describes the roagation of the sound beams in the homogeneous and thermoviscous (heatconducting fluids. The advantage of the model is that it combines the effects of diffraction, absortion and nonlinearity. Even though the KZK equation is esecially suitable for the far field of the araxial region for moderately focused or unfocused transducers, its original and revised form has also been alied to more general ultrasound simulations. Due to the broad alications of the KZK equation, many numerical methods have been develoed. In 984, Aanonsen et al. [] develoed numerical algorithms to solve the KZK equation in the frequency domain. In articular, the authors alied the Fourier exansions to the original equation, which leads to a couled PDE system and they further solved the resulting system using finite difference method. Bakhvalov et al. [3] roosed a mixeddomain numerical method which solves the diffraction and absortion terms in the frequency domain, and comutes the nonlinear term using the Godunov method in the time domain. Later, Lee and Hamilton [5] designed an oerator-slitting finite difference method to solve the KZK equation entirely in the time domain; for each udate along the direction of the wave roagation, they solved the diffraction term, absortion term and nonlinear term sequentially. In order to handle the oscillatons caused by the diffraction term, the authors emloyed the imlicit backward finite difference method in the near field, and the Crank Nicolson method in the far field. The stretched coordinates method [4] was used to deal with the nonlinear term of the equation. Many other algorithms including the work by Baker et al. [], Yang et al. [3] and Hajihasani et al. [] were then develoed based on Lee and Hamilton s framework to tackle more general situations. For examle, Yang et al. and Hajihasani et al. consider the diffraction term in Cartesian coordinates. Since the diffraction term is the main challenging art in the numerical simulations, they rovide different treatments in order to solve the equation accurately. Generally seaking, most of the numerical schemes of the KZK equation are based on either the sectral method or finite difference methods. In this aer, we roose a local discontinuous Galerkin (LDG method to solve the KZK equation. Discontinuous Galerkin (DG methods, which use iecewise defined functions (tyically olynomials in each element, were initiated by Reed and Hill s work [6] to solve the steady state linear hyerbolic equations. DG methods have drawn a great amount of attention since the ioneering works by Cockburn and Shu in a series of aers [5,7 9]. In articular, the authors established a framework to solve nonlinear hyerbolic conservation laws by introducing total variation bounded nonlinear limiters [7] to cature the strong shocks. More details for the develoment of the DG methods can be referred to the review aers [6,]. The DG methods were then develoed to solve convection-diffusion equations and artial differential equations with high order satial derivatives. For instance, Yan et al. used the LDG method to solve a general Korteweg de Vries (KdV tye equation in [], and they further generalized the LDG method for equations with fourth and fifth order satial derivatives in []. Later, Xu and Shu alied the LDG method to solve many other equations including the Kadomtsev Petviashvili and Zakharov Kuznetsov equations [9], and we refer to their review aer [] for the latest develoment of the LDG method. The reason we design a LDG method to solve the KZK equation is that the DG methods have many advantages. The DG methods can be easily formulated to be of arbitrary order of accuracy, and they are also suitable for adative simulations in both the - and h-refinement,

3 J Sci Comut (7 73: comlicated geometry and arallel comuting. Based on the aforementioned benefits, we roose a LDG method to solve the KZK equation in the time domain so that it is easy to deal with the non-local diffraction term and the diffusion term. To the best of our knowledge, this aer rovides the first algorithm to the KZK equation that belongs to the finite element schemes. One feature of the KZK equation is that the solutions are much more oscillatory in the near field comared to the solutions in the far field. Such roerty leads to different choices in terms of discretization within distinct regimes of the domain. That is, we use two different ste sizes along the direction of the beam roagation (in the σ domain of Eq. (.8 for all the examles excet for the accuracy test in Sect Another secial treatment is that we multily the KZK equation by the cylindrical radial variable (ρ before we derive the weak formulation of the LDG scheme. That gives the rovable L stability of our LDG scheme for the KZK equation in the cylindrical coordinates. For our numerical simulations, we mainly comare the erformance of our numerical results with Lee s results [4] for some benchmark examles. In general, the examles fall into the two categories, i.e., the cases for the focused and unfocused sources. We observe that our numerical results match well with Lee s results for the unfocused case, and have better accuracy for the focused case. The outline of our aer is as follows. In Sect., we introduce two dimensionless forms of the KZK equation, roose LDG scheme for the KZK equation and resent the stability analysis of the scheme. In Sect. 3, we erform some accuracy tests of our scheme on the KZK equation with only the diffraction term or with an extra source term. We also test our scheme for some classical numerical examles including the focused circular short tone burst excitation and the roagation of unfocused sound beams. By comaring our results to the benchmark solutions, we can show that under the same degrees of freedom, our scheme has better erformance. We conclude this aer in Sect. 4. Local Discontinuous Galerkin Discretization. Model Problem The original form of the KZK equation for the axisymmetric sound ressure in cylindrical coordinates is given by z t = c ( r + + D 3 r r c 3 t 3 + β ρ c 3, (. t Here (r, z reresents the cylindrical coordinates with z denoting the axis of the beam which is the direction of roagation and r being the radial distance from the z axis. Another variable t is the retarded time which could be negative. Other constant arameters c, D,β and ρ reresent the sound seed, diffusivity of the sound in the thermoviscous fluid, nonlinearity arameter and the density of the fluid, resectively. In this aer, we consider two dimensionless forms of the original KZK equation (.. In order to model the focused source, one can use the variable transformations = /,σ = z/d,ρ = r/a, = ω t with being the amlitude of the sound ressure for the source, d being the focal length of the source, a being the radius of the source and ω being the frequency of the source, to obtain the following non-dimensional KZK equation σ = 4G (ρ ρ ρ d + A + N ρ (, (.

4 596 J Sci Comut (7 73: where G, A and N are constants which satisfy G > anda, N. If we instead aly the following transformations = ( + σ/,σ = z/z,ρ = r/a/( + σ, = ω t (r/a /( + σto (., where z is the so-called Rayleigh distance, we can obtain the case for unfocused sources: σ = 4( + σ (ρ ρ ρ d + A + N ρ + σ (. (.3 The major difference of the two dimensionless forms is essentially how the relative range coordinate σ is defined. More details about these transformations can be referred to [4]. For both Eqs. (. and(.3, the terms on the right-hand side denote diffraction, absortion and nonlinearity, resectively. The source condition for a weakly focused, axisymmetric beam can usually be defined by = f (t + r /c dg(r, at z =, (.4 where the function f is the temoral excitation and g(r is the amlitude distribution. In the dimensionless coordinates, this source condition becomes: = f ( + Gρ g(ρ, at σ =. (.5 We define the temoral and sacial domain as = ρ min ρ ρ max and min max, and imose the following boundary conditions: ( min,ρ,σ=, ( max,ρ,σ=, (, ρ max,σ=, ρ (, ρ min,σ=. (.6 The satial domain must be large enough to minimize the error introduced by the inevitable reflection due to the artificial boundary conditions. The time domain must also be large enough to encomass both the axial wave and the delayed edge wave. Lee has a detailed discussion on how to select ρ max, min and max in his revious work [4]. Here, we resent his result for the focused sources: ρ max = 4, min = (G + ω T + π, max = ω T + π, (.7 where T is the duration of the source excitation. We will secify the domain in each numerical examle accordingly. The equation for the focused case (. will be used in the following subsections to derive our LDGmethod. Due to the setu of our roblems, although is the actual temoral variable, we follow the aroach in [4], and regard σ as the artificial temoral direction, ρ and as satial variables when we solve the equations numerically.. Notations We consider a two-dimensional rectangular domain and without loss of generality, we denote it by [ min, max ] [ρ min,ρ max ]. We discretize the comutational domain into rectangular cells K i, j = I i J j,wherei i =[ i, i+ ],fori =,...,N and J j = [ρ j,ρ j+ ],for j =,...,N ρ. The coordinate for the center of the cell K i, j is denoted by ( ( i,ρ j := ( i + i+ /,(ρ j + ρ j+ /, and the mesh sizes in -andρ-direction are denoted by i = i+ i and ρ j = ρ j+ ρ j, resectively. We define the iecewise olynomial sace as Vh k ={v : v K i, j Q k (I i J j, K i, j },whereq k is the sace of tensor roducts of one-dimensional olynomials of degree u to k. Note that functions in

5 J Sci Comut (7 73: V k h can be discontinuous across element boundaries. For any function h V k h, h( + i+/,ρ and h ( i+/,ρdenote the limit values of h at i+ from the right cell I i+ J j and from the left cell I i J j, resectively. Similar definitions hold for h (, ρ + j+/ and h(, ρ j+/..3 LDG Method for the KZK Equation In this subsection, we define the semi-discrete LDG method for the KZK equation (.. That is, we only discretize the domain of ρ and, and regard σ as the continuous time. The KZK equation is written into a first order system by substituting the first order derivatives ρ, and definite integral min ρ d with the auxiliary variables s, q, r, resectively: σ = 4Gρ (ρr ρ + Aq + N (, (.8 s = ρ, (.9 r = s, (. q =, (. where s = sd. (. min Our scheme for (.8 (. can be formulated as follows: we look for h, s h, q h, r h Vh k, such that d h φρdρd = ( r h φ ρ(,ρ dσ K ij 4G j+ r h φ + ρ(,ρ I j d r h φ ρ ρdρd i 4G K ij + A ( q h φ ρ( i+,ρ q h φ + ρ( J i,ρdρ A q h φ ρdρd j K ij + N ( h ρ( i+,ρ J h φ+ ρ( i,ρdρ N h j ρdρd, K ij (.3 s h ψρdρd = ( h ψ ρ(,ρ j+ h ψ + ρ(,ρ K ij I j d i h (ψρ ρ dρd, (.4 K ij r h ϕρdρd = s h ϕρdρd, (.5 K ij K ij q h ζρdρd = ( h ζ ρ( i+,ρ h ζ + ρ( K ij J i,ρdρ j h ζ ρdρd, (.6 K ij hold for all test functions φ, ψ, ϕ, ζ Vh k. Note that we multily the equations by ρ and then integrate them, since the roblem is axisymmetric and ρ denotes the radial distance. The terms, h, q h, h, r h and h in (.3 (.6 are the so-called numerical fluxes resulting from integration by arts. They must be designed carefully at the boundary and they are essential to ensure numerical stability. For the fluxes h, q h, h and r h,weusethe simle alternating fluxes,

6 598 J Sci Comut (7 73: h = + h, q h = q h, h = + h, r h = r h, (.7 where all quantities are comuted at the same oints (i.e., the cell interface. We remark that the choice of the fluxes (.7 is not unique. We can, for examle, alternatively choose the numerical fluxes to be h = h, q h = q + h, h = h, r h = r + h. (.8 For the flux h, we use the simle Lax Friedrichs flux defined by h (h, + h = ( ( h + ( h + + α( h + h, α = max h, (.9 where the sign before α is ositive as the nonlinear term aears on the right hand side of the equation. The following are the details of imlementation of our roosed LDG methods. The first non-trivial comonent comes from the non-local term s h in (.5. As defined in (., this term can be calculated exactly since s h is now a iecewise olynomial in. The other issue comes from the integration in olar coordinate ρ. The mass matrix now deends on the location of ρ, and may be different in cell K ij for different j. Therefore, we re-calculate these mass matrices at the beginning, and use them in the udate of each time ste. In order to imlement the boundary conditions, we aly the Neumann boundary condition at ρ min and set r h (, ρ min =. (. At ρ max, min, max, when Dirichlet boundary conditions are alied, we set: r h (, ρ max = r h (, ρ max, h (, ρ max = (, ρ max =, (. q h ( max,ρ = q h ( max,ρ, h ( max,ρ= ( max,ρ=, (. and q h ( min,ρ= q h + ( min,ρ, h ( min,ρ= ( min,ρ=. (.3 Note that (.3 indicates that the directions of q h and h are flied at the left boundary min. Alying the semi-discrete LDG scheme (.3 (.6 to the KZK equation, we end u with an ordinary differential equation system which can be written as u σ = L(u, (.4 where L(u is the discretization of the oerators on the right hand side of KZK equation. To udate u, we choose a class of high order total variation diminishing (TVD Runge Kutta (RK tye methods develoed by Shu and Osher [8]. In articular, we consider the following second-order TVD Runge Kutta method: u n, = u n + σ n L(u n, u n+ = un + un, + σ n L(u n,, to udate the numerical solution from σ n to σ n+, with σ n = σ n+ σ n.

7 J Sci Comut (7 73: L Stability In this subsection, we first show that the quantity ρ dρd, thel norm of in the cylindrical coordinate, is a non-increasing function in the σ -direction for the exact solution of the KZK equation. Again, the extra ρ is included in the definition of L norm, as the roblem is axisymmetric and ρ denotes the radial distance. Then we rove that ρ h dρd of the numerical solution h is also non-increasing. This roerty indicates that our numerical scheme is stable. Proosition The solution to the KZK equation (. along with boundary conditions (.6 satisfies: d ρ dρd. (.5 dσ Proof We multily the Eq. (.byρ, integrate over the domain to obtain d ρdρd = dσ 4Gρ (ρ ρ ρ ρdρd + A ρdρd + N ( ρdρd = max (ρ ρ ρ max ρ 4G min d (( ρ ρdρd min 8G ρmax + A ρ max min dρ A ρ min ( ρdρd + N ( 3 ρdρd 3 = (( ρ ρdρd A ( ρdρd 8G = ρmax ( max ρ d ρdρ A ( ρdρd 8G ρ min min, where the boundary conditions ρ (, ρ min,σ =, (, ρ max,σ =, ( min,ρ,σ = and ( max,ρ,σ= areused. We then show that the roosed semi-discrete LDG method will result in a numerical solution with similar roerty, that is, the scheme is L -stable. Proosition (cell entroy inequality There exist numerical entroy fluxes H i, j+,i i+, j and J i+, j such that the solution to the scheme (.3 (.6 satisfies: ( h σ h ρdρd + (H i, j+ H K i, j + (J i+, j J i, j + (I i+, j I i, j ij + 4G K ij ( s h ρdρd. (.6 Proof Taking the test functions φ = h, ψ = r h, ϕ = s h, ζ = q h in Eqs. (.3 (.6, we have: K ij ( h σ h ρdρd = 4G I i r h h ρ(,ρ j+ r h h + ρ(,ρ j d r h ( h ρ ρdρd 4G K ij

8 6 J Sci Comut (7 73: A q h h ρ( i+,ρ q h + J h ρ( i,ρdρ A q h ( h ρdρ j K ij + N h h ρ( i+,ρ J h + h ρ( i,ρdρ N h j ( h ρdρd, K ij (.7 s h r h ρdρd = h rh ρ(,ρ j+ h r + K ij I h ρ(,ρ j d h (r h ρ ρ dρd, (.8 i K ij r h s h ρdρd = s h s h ρdρd, (.9 K ij K ij q h q h ρdρd = h qh ρ( i+,ρ h q + K ij J h ρ( i,ρdρ h (q h ρdρd. (.3 j K ij We choose the alternating flux (.7 along with (. (.3, multily Eqs. (.8 and(.9 by /(4G, multily Eq. (.3 bya, and sum them u with (.7. After some algebraic maniulations, we have: ( h σ h ρdρd + A qh ρdρd + K ij K ij 4G + (H i, j+ H i, j + (J i+, j J i, j + where the entroy fluxes are: H i, j+ = 4G J i+, j = A I i+, j = N J j with the excetion at the boundaries: H i, = 4G J, j = A and the extra i, j term is defined by i, j = N K ij rh + h ρ(,ρ j+ d, I i qh + h ρ( i+,ρdρ, J j J j h ( i+,ρ ( h h rh + h ρ(,ρ d, I i q h + h ρ(,ρdρ, J j + h ( i,ρ h ( i,ρ By the monotonicity of h,wehave. ( s h ρdρd (I i+, j I i, j + i, j =, (.3 d h ρdρ, ( h h d h ρdρ. Notice also that A qh ρdρd. K ij Therefore, we can derive (.6 and comlete the roof.

9 J Sci Comut (7 73: The L stability of the numerical method is a corollary as follows. Proosition 3 The numerical solution to the scheme (.3 (.6 satisfies: d ρh dρd. (.3 dσ Proof We sum u (.6 over i and j and obtain N ( N ρ ( h σ h ρdρd + H i,nρ + H i, + (J N +, j J, j j= i= N + (I N +, j I, j + 4G j= ( s h ρdρd. Following the definition of fluxes at the boundary (. (.3, we can obtain We also have I N +, j = N = N = + N, H i,nρ + = H i, = J N +, j = J, j =. h ( N + ( h h d h + h ( N + ( h h d h + N ( + h N + h ( N + ( h h d h ( + h N + h ( N + ( h h d h I, j =, and 4G ( s h ρdρd = ρmax ( max s h d ρdρ. 4G ρ min min Finally we can obtain d ρh dρd. dσ Remark In the case when the boundary conditions (.6 are changed to eriodic boundary conditions, the stability inequalities (.5, (.6and(.3 still hold. 3 Numerical Exeriments In this section, we imlement the reviously described LDG algorithm together with an exlicit second order TVD Runge Kutta integration. We will show the accuracy tests to verify the rate of convergence, and also examine the erformance of our scheme on the KZK

10 6 J Sci Comut (7 73: equation with the diffraction term only, for which exact solutions exist. Full KZK equation will be simulated and the erformance of our LDG methods will also be comared with the results of the finite difference methods in Lee [4]. Here let us briefly summarize Lee s method first. Lee used central difference method to aroximate the satial derivatives and traezoidal rule to aroximate the integral. In the near field along the roagation of the wave, he used imlicit backward Euler method to udate the solution, and used Crank Nicolson method in the far-field. 3. Examle : KZK Equation with the Diffraction Term Only 3.. Accuracy Test We start by considering the KZK equation with the diffraction term only, which contains both the derivative and integral. In this examle, we erform a convergence test to check the accuracy of our algorithm. By manufacturing an exact solution, we can construct a source term to be added to the equation. The equation, after we add an extra term h(,ρ,σ, becomes: σ = (ρ ρ ρ d + h(,ρ,σ, π ρ 3π, π, (3. 4Gρ min where ( h(,ρ,σ=e σ sin( + ρ (sin( + ρ sin(ρ + ρ(cos( + ρ cos(ρ. 4Gρ We assume the eriodic boundary conditions for and ρ: and source condition at σ = : (,π,σ= (, 3π, σ, (,ρ,σ= (π, ρ, σ, (3. Therefore, this roblem has the exact solution of the form In the simulation, we use the following arameters: (, ρ, = sin( + ρ. (3.3 (,ρ,σ= e σ sin( + ρ. (3.4 σ end =., σ = 4 3, (3.5 where σ end is the final stoing oint and σ is the initial ste size for the case N =, and we choose G =. The mesh size is halved when we double N. We erform the accuracy test for both Q and Q saces. The first order accuracy for Q sace and second order accuracy for Q sace are exected. Error Tables and clearly show that the exected accuracy are achieved for the unknown h in both cases. We observed that the convergence rate of s h is.5 for Q sace for the equation with the diffraction term only. Various numerical exeriments show that, if the diffusion term is included, the convergence rate of s h will imrove, which can also be seen in Table Linear Lossless KZK Equation Next, let s consider the linear lossless KZK equation, which contains only the diffraction term, with a focused circular source of uniform amlitude distributions, where the exact

11 J Sci Comut (7 73: Table Examle with Q sace: numerical errors and orders with uniform meshes N L error Error of Order Error of s Order Error of r Order E E E E. 3.53E.6.756E E. N L error Error of Order Error of s Order Error of r Order 5.6E 9.338E E E E E E E E E.3.538E E E E.6 Table Examle with Q sace: numerical errors and orders with uniform meshes N L error Error of Order Error of s Order Error of r Order.884E E E E E E E E E E E E E 4.6 N L error Error of Order Error of s Order Error of r Order 4.55E 4.8E 6.97E E E E E E E E E E E E E solution in the axial wave forms exists. Consider a Gaussian modulated sinusoid source wave form with radius a in (.4 by secifying f and g: [ ( ] ω t f (t = ex sin(ω t, (3.6 3

12 64 J Sci Comut (7 73: and {, r a, g(r =, r a. In dimensionless arameters, this source condition becomes [ ( + Gρ ] ex sin ( + Gρ, ρ, (, ρ, = 3, ρ. The boundary condition (.6 isused. As shown in [,3], the exact axial solution (at ρ = takes the form of ( (, ρ =,σ= ex σ sin ( + G G σ ( ( sin( ex 3. In our simulation, we used the following arameters: ( 9 ( + G G σ G =, min = π, max = 4π, =., ρ max = 4, ρ =., σ = 5 5, σ = 4, (3.7 where σ is used for σ., and σ is used for σ>.. In the far-field when σ>., the ressure usually varies smoothly, which allows us to choose a larger ste size. We comare the erformance of our LDG method with Lee s algorithm. For fair comarison, when we use LDG method with Q sace, we choose the satial mesh that is two times less dense than Lee s to ensure that both algorithms have the same degree of freedom. For LDG method with Q sace, we use the same satial mesh size as in Lee s method. The time evolution lots from different schemes are overlaid and comared with the exact solution in Fig.. Some oscillations can be seen when σ is small, which are caused by the discontinuity in the source condition. At σ =., LDG method in both Q and Q sace roduces oscillated waves coming from the right. Notice here the horizontal axis reresents the dimensionless retarded time. Note that Lee s algorithm has no visible oscillated waves on the right since it used the imlicit backward temoral method which is effective in suressing oscillations. The visual effect that two major waves traveling closer as σ becomes greater agrees with the fact that the difference of the arrival time of the axial wave and edge wave diminishes in the far-field. If we concentrate on the magnitude of the edge wave sike (the right sike for σ =.,.3,, we can observe that the LDG scheme in Q sace resulted in a much better aroximation, while the result from Lee s method only becomes comarable when σ. A zoomed-in view at the ressure at σ = is shown in Fig., where we can observe that the difference between results of LDG with Q sace and the exact solution is almost indistinguishable at the exhibited measure, while discreancy is more obvious between the exact solution and LDG scheme with Q sace and Lee s finite difference method. 3. Examle : Accuracy Test for the Full KZK Equation Next, we consider the full KZK equation with all three terms on the right hand side. To check the accuracy of the roosed methods, we again manufacture an exact solution, and, as a result, add an extra source term to the KZK equation. We consider the following equation:

13 J Sci Comut (7 73: σ =. LDG Q LDG Q Exact.5 σ =. LDG Q LDG Q Exact σ =.3 LDG Q LDG Q Exact.5 σ = LDG Q LDG Q Exact σ =.7 LDG Q LDG Q Exact σ = LDG Q LDG Q Exact Fig. Comarison of LDG, Lee s method and exact solution for axial wave forms for a focused uniform iston source with a Gaussian modulated ulse, at σ =.,.,.3,,.7,

14 66 J Sci Comut (7 73: Fig. Zoomed-in lot for Fig. at σ =.7.68 σ = LDG Q LDG Q Exact σ = (ρ ρ ρ d + A + N 4Gρ min ( +h(,ρ,σ, π ρ 3π, π, (3.8 where the extra source term is given by h(,ρ,σ= e σ N ( sin( + ρ + e σ sin( + ρ (sin( + ρ 4Gρ sin(ρ + ρ(cos( + ρ cos(ρ + A sin( + ρ. (3.9 We again assume the eriodic boundary conditions as in (3., and the source conditions in (3.3. This roblem has the same exact solution as in (3.4. The arameters in (3.5 are also used here. The error tables are dislayed in Tables 3 and 4, where we obtain the first order accuracy with Q sace and second order accuracy with Q sace as exected. 3.3 KZK Equation with Focused Source In this examle, we investigate the roagation of a non-linear ultrasound beam generated by a focused circular short tone bursts (ulses with only a few cycles. Consider the wave form (.5 with f (t given by [ ( ] ω t f (t = ex sin(ω t. (3. 3π and g(r defined in (3.7. Written with resect to the dimensionless arameters, the source form becomes [ ( + Gρ ] (, ρ, = ex sin( + Gρ. 3π

15 J Sci Comut (7 73: Table 3 Examle with Q sace: numerical errors and orders with uniform meshes N L error Error of Order Error of s Order Error of q Order Error of r Order E E E E. 7.67E E E.4.777E E.4 N L error Error of Order Error of s Order Error of q Order Error of r Order E 9.539E 9.44E E E E E E E.46.88E E E E.8.395E E E. 5.89E E E.3

16 68 J Sci Comut (7 73: Table 4 Examle with Q sace: numerical errors and orders with uniform meshes N L error Error of Order Error of s Order Error of q Order Error of r Order.88E E E E..5E E E E E E E E E E E E E E N L error Error of Order Error of s Order Error of q Order Error of r Order 4.83E 3.498E.973E 3.658E E E E E E E E E E E E E E E E E.77

17 J Sci Comut (7 73: σ =. LDG Q.5 σ =.3 LDG Q σ =.7 LDG Q 8 σ =.8 LDG Q σ =.5 LDG Q.5 σ = LDG Q Fig. 3 Comarison of axial wave forms of KZK equation with focused source simulations from Lee s algorithm and LDG method with Q sace

18 6 J Sci Comut (7 73: Fig. 4 Comarison of axial wave forms of KZK equation with focused sources simulations at σ =. using refined mesh. ρ =., =.5, σ = 3 5 are used for LDG method. ρ =./3, =.5/3, σ = 5 are used for Lee s method.5 σ =. LDG Q We imose the boundary condition (.6. The arameters used in this examle are A =., N =, G = 5, ρ max = 4, min = 5π, max = 3π, ρ =., =., σ = 5 5. These settings are in accordance with the original setting in (.7 from [4]. Again, we use a mesh size that is half of that in the simulation of finite difference algorithm, in order to rovide a fair comarison with Lee s results. In Fig. 3, it can be observed that at σ =., both the solutions of LDG and Lee s method have a wave coming from the right boundary. That is similar to the roagation of the edge wave. However, the solution of the finite difference algorithm has much smaller magnitude for such wave than that of our LDG method. As the wave roagates, the central wave and the edge wave merge. The maximal ositive ressure aeared at σ =.7 in Lee s solution, while with LDG method, this aears at σ =.8. The solutions of the LDG scheme and the finite difference scheme agree relatively well afterwards. We can clearly observe from Fig. 3 that the major difference between the two solutions occurs at σ =.. In order to determine which one is more accurate, we refine the mesh for these two methods and comare the solutions. As we decrease ρ, and σ, the magnitude of the edge wave for both methods increases. When the mesh size is small enough, the two solutions eventually aroach each other. Figure 4 shows the solution by our LDG method with ρ =., =.5 and σ = 3 5, which is about half the size of the arameters in Fig. 3. In contrast, for the Lee s method, we have to use three times finer mesh size than that of LDG method to achieve similar solution. Such observations indicate that our LDG method is more efficient than Lee s method for this numerical examle. In Fig. 5, we dislay the three-dimensional view of waveforms at different σ.

19 J Sci Comut (7 73: Fig. 5 Three-dimensional lot of the evolution of a focused iston source 3.4 Unfocused Pulsed Sound Beams In this last examle, we investigate the roagation of unfocused, ulsed, finite amlitude sound beams in the thermoviscous fluids. The LDG algorithm can deal with any

20 6 J Sci Comut (7 73: σ = LDG Q.8.6 σ = LDG Q σ = 3.8 LDG Q σ = 5 LDG Q σ = 9 LDG Q -3 σ = 3 LDG Q Fig. 6 Comarison of axial wave forms of KZK equation with unfocused source simulations from Lee s algorithm and LDG method with Q sace

21 J Sci Comut (7 73: σ = 9, ρ = LDG Q -3.5 σ = 9, ρ =.48 LDG Q σ = 9, ρ =.99 LDG Q -3 σ = 9, ρ =.984 LDG Q σ = 9, ρ = 3. LDG Q σ = 9, ρ = 4 LDG Q Fig. 7 Comarison of wave forms of KZK equation with unfocused source simulations from Lee s algorithm and LDG method with Q sace when σ = 9

22 64 J Sci Comut (7 73: σ = 3, ρ = LDG Q -3 σ = 3, ρ =.48 LDG Q σ = 3, ρ =.99 LDG Q -3 σ = 3, ρ =.984 LDG Q σ = 3, ρ = 3. LDG Q -4 3 σ = 3, ρ = 4 LDG Q Fig. 8 Comarison of wave forms of KZK equation with unfocused source simulations from Lee s algorithm and LDG method with Q sace when σ = 3

23 J Sci Comut (7 73: axisymmetric sound sources, that is, the amlitude distribution g(r in the source condition can be arbitrary under the constraints imosed by the arabolic aroximation. For simlicity, here we consider the source with a uniform amlitude evolution. The equation for the unfocused source is given by (.8. The algorithm (.3 (.6can be alied with a slight modification on the coefficients. We use the ste and mesh sizes suggested by Lee [4]. The source waveforms defined in (.5 and(3.7 are considered in this examle. Written with resect to the dimensionless arameters after the variable transformation, the source becomes: ( ( ( + ρ ex sin ( + ρ, ρ <, (, ρ, = 3π (3., ρ. We use (.6 as the boundary conditions. The following arameters are used in this examle: A =, N =, ρ max = 8, min = 3π, max = 3π, ρ =.3, =., σ = 5 5. The comarison of the results of LDG methods and Lee s finite difference algorithm is shown in Figs. 6, 7, and8. We can observe that these two methods rovide similar results both on and off axis. 4 Concluding Remarks In this aer, a LDG method was roosed for solving the KZK equation, which is widely used in medical ultrasound simulations. L stability analysis of our LDG method has been rovided. Because the solutions of the KZK equation have stronger oscillations in the near field (with relatively small σ than in the far field (with large σ, we choose smaller ste size for σ. and larger ste size elsewhere in our simulations. Such choice is good to cature the correct magnitude of the oscillations and kee the simulations efficient. In the future, time integrator such as imlicit temoral method will be imlemented to further accelerate the calculation. Based on our numerical results, we observe that our scheme rovides more accurate results for the case of a focused uniform iston source with a Gaussian modulated ulse, when comared with the solution by an existing finite difference aroach. For the unfocused ulsed sound beams, our numerical solutions match well with the benchmark solutions. These numerical results indicate that our LDG scheme has great otential in various alications of the KZK equation. References. Aanonsen, S.I., Barkve, T., Tjotta, J.N., Tjotta, S.: Distortion and harmonic generation in the nearfield of a finite amlitude sound beam. J. Acoust. Soc. Am. 75, (984. Baker, A.C., Berg, A., Sahin, A., Tjotta, J.N.: The nonlinear ressure field of lane rectangular aertures: exerimental and theoretical results. J. Acoust. Soc. Am 97, ( Bakhvalov, N.S., Zhileikin, Y.M., Zabolotskaya, S.A.: Nonlinear Theory of Sound Beams. American Institute of Physics, New York ( Blackstock, D.T.: Connection between the Fay and Fubini solutions for lane sound waves of finite amlitude. J. Acoust. Soc. Am. 39, 9 6 (966

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