Hyperbolic Method for Dispersive PDEs: Same High- Order of Accuracy for Solution, Gradient, and Hessian

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1 AIAA Aviation 3-7 Jne 6, Washington, D.C. 46th AIAA Flid Dynamics Conference AIAA Hyperbolic Method for Dispersive PDEs: Same High- Order of Accracy for, Gradient, and Hessian Downloaded by NASA LANGLEY RESEARCH CENTRE on Agst 4, 6 DOI:.54/ Alireza Mazaheri NASA Langley Research Center, Hampton, VA 368 Mario Ricchito INRIA Bordea Sd-Oest and Institt de Mathématiqes de Bordea, 3345 Talence Cede, France Hiroaki Nishikawa National Institte of Aerospace, Hampton, VA 3666 In this paper, we introdce a new hyperbolic first-order system for general dispersive partial differential eqations (PDEs). We then etend the proposed system to general advection-diffsion-dispersion PDEs. We apply the forth-order RD scheme of Ref. to the proposed hyperbolic system, and solve time-dependent dispersive eqations, inclding the classical two-soliton KdV and a dispersive shock case. We demonstrate that the predicted reslts, inclding the gradient and Hessian (second derivative), are in a very good agreement with the eact soltions. We then show that the RD scheme applied to the proposed system accrately captres dispersive shocks withot nmerical oscillations. We also verify that the soltion, gradient and Hessian are predicted with eqal order of accracy. I. Introdction Dispersion effects play a fndamental role in many applications involving hydrodynamics. At the large scale, the flow is dominated by advection, while the dissipative effects are more important at the microscopic level. At the mesoscopic level (the intermediate level) the dispersive effects become important, as it is the case, for eample, in nonlinear optics, electromagnetism, qantm mechanics, 4 relativity and Bose-Einstein condensates, 5, 6 atmospheric, coastal, and flvial hydrodynamics, 7 magma, highly viscos flids and/or capillary effects. 4 The main physical effects associated with dispersion are the appearance of dispersive (or ndlating) shocks, and the eistence of smooth traveling solitary waves, which may prodce comple interactions with one another. These systems, independent of the physical natre of the involved medim, admit a mesoscale hydrodynamic model, which consists of a set of Partial Differential Eqations (PDEs). These PDEs are mostly reglarizations of hyperbolic models that, in one-dimension, may be written as t + F () = D, () where, in a classical sense (endowed with an entropy pair, with a diagonalizable fl Jacobian F (), etc.), the left hand side defines a hyperbolic model. Depending on the application and the physical hypotheses made, the reglarization on the right hand side may take different forms, and have a dissipative and/or dispersive character (see e.g., Refs. 3, 5 9). Therefore, term D, may be a combination of the following forms: (a) dissipative reglarization D = ν, (b) dispersive reglarization with time derivative D = ɛ t, Research Aerospace Engineer, Aerothermodynamics Branch, M/S 48A, alireza.mazaheri@nasa.gov Grop Leader at INRIA Sd-Oest, mario.ricchito@inria.fr Associate Research Fellow, hiro@nianet.org of 3 This material is declared a work of the U.S. Government and is not sbject to copyright protection in the United States.

2 (c) flly dispersive reglarization D = ɛ. The form (a) is a classical viscos reglarization, while form (b) can be recast as a first-order evoltionary PDE with an embedded steady state second-order time independent elliptic problem; i.e., t w = f, + = w. Downloaded by NASA LANGLEY RESEARCH CENTRE on Agst 4, 6 DOI:.54/ We previosly showed e.g., in Refs.,, how one can constrct very accrate nmerical approimations of problems of form (a) and the elliptic form of (b), by first reformlating the system of PDEs as a first-order hyperbolic system. The schemes proposed with the hyperbolic formlation of PDE systems are pwind and highly accrate for both the soltion and its gradient, and have a natral potential for etension on arbitrary nstrctred meshes as illstrated in Refs., 3. The presence of a third-order derivative term in form (c), however, introdces discretization difficlties, which are often related to the nderstanding of the type of stencil that is reqired to approimate these high-order derivative terms, the stability of the method sed, and the imposition of bondary conditions. In Refs. 4 and 7, possible soltions to some of these isses are proposed, where the athors showed very good reslts with a non-hyperbolic first-order system reformlation of a PDE and carefl discretization of the fles. The hyperbolic reformlation of dispersive PDEs similar to the one initially proposed for diffsion in Ref. alleviate the above mentioned isses. However, it is shown and proved in Ref. 5 that the hyperbolic formlation of a dispersive PDE in the form given in Ref. is not possible. Ths, we are motivated to introdce an alternative hyperbolic formlation that is careflly designed for general dispersive PDEs that are relevant to, for eample, qantm mechanics, relativistic hydrodynamics, and coastal engineering 5, 8, 4 applications, sch as the Korteweg-de Vries (KdV) eqation. In this work, we first present a first-order system for a pre dispersion eqation, and show how the proposed formlation can be made hyperbolic. As an intermediate generalization, we then show that an advective-dispersive PDE (sch as the classical KdV) can also be made flly-hyperbolic as well. The flly-hyperbolic advection-dispersion system cold be sefl in imposing a characteristics bondary condition. A practical etension of the proposed hyperbolic dispersion system for general advective-diffsivedispersive PDEs follows net. We then present some nmerical eamples by applying the high-order residaldistribtion (RD) scheme of Ref. to the proposed system, and solving general dispersive PDEs, inclding the classical KdV eqation, on randomly distribted nodes. We verify the order of accracy of the scheme for both the soltion, the gradient, and the Hessian (second derivative) with the se of method of manfactred soltion, and show that the RD scheme applied to the proposed hyperbolic system prodces accrate soltion, gradient and Hessian with eqal order of accracy. The ability to obtain accrate gradient and Hessian are very important as they are sed in many hydrodynamic dispersive models to define physically relevant qantities (e.g., potentials, energy, flow properties at an arbitrary depth, etc.). We also present soltions for the zero dispersion limit of conservation laws and demonstrate that the RD scheme applied to the proposed hyperbolic system is robst and can captre physical oscillations associated with the generated dispersive shocks. II. Hyperbolic Dispersion In this section, we start with a time-dependent dispersive PDE, and reformlate it to a first-order system that can be sccessflly transformed to a first-order hyperbolic system. Consider the following linear dispersive PDE that is often referred to as the Airy eqation, t = ɛ, () where ɛ is the dispersion coefficient (positive or negative). Following the process we otlined in Ref. (althogh other choices may also be possible), we consider here the semi-discrete form of Eq. () obtained with some implicit time integration scheme: α t = ɛ + s(),, where α and s() depend on the time discretization and the known vales of. We then replace the of 3

3 semi-discrete PDE by the steady limit of the psedo-time dependent system τ = ɛ q α t + s(), τ p = T ɛ ( p γ), (3) τ q = T ɛ (γ + p q), Downloaded by NASA LANGLEY RESEARCH CENTRE on Agst 4, 6 DOI:.54/ where τ is the psedo time, t is the physical time, and T ɛ and γ are, respectively, the dispersion relaation time and an arbitrary constant, both to be defined later. It is easy to verify that in the above system satisfies the original semi-discrete dispersion eqation at psedo steady state (see also Refs. and ). Note that, the proposed system with γ = redces to the first-order system formlation that is proven in Ref. 5 not to be hyperbolic. The stdy of the characteristic polynomial of the Jacobian matri of the qasi-linear differential operator on the right hand side of Eq. (3) reveals that the form of γ is critical in constrcting a hyperbolic system. For eample, we have fond that the following form of γ cold reslt in a hyperbolic formlation of the first-order system for dispersion γ = β + ɛ β T ɛ, (4) where for dimensional consistency β = κ/l ɛ, κ is an arbitrary constant, and L ɛ is the dispersion length scale, to be defined later. With this choice of γ, the first-order system (3) admits the characteristic speeds = βt ɛ, = λdisp ( + + 4ɛβ 3 T ɛ ), 3 = λdisp with the following corresponding eigenvectors ɛ T ɛ 3 T ɛ R = ɛβ β /(ɛβ) β 3 /(ɛβ) ( + 4ɛβ 3 T ɛ ), (5), (6) which is non-singlar (nless κ =,, or ). We now define the relaation time T ɛ as the ratio between the dispersion length scale, L ɛ, and either of the λ or the λ 3, and arrive at the following relation: T ɛ = L3 ɛ ɛ, (7) where we have chosen κ = κ g (+ 5)/, which is denoted here as the golden ratio, satisfying κ g κ g = by definition. We remark that the proposed system have three real eigenvales with linearly independent eigenvectors for any positive κ (ecept for κ =, which makes the matri of eigenvectors singlar). Ths, the proposed first order system is hyperbolic. With the choice of the golden ratio, we arrive at γ = /L ɛ. With the definitions of T ɛ, κ(= κ g ), β, and γ, we recast the proposed hyperbolic dispersion system in a vector form as τ U + A U = Q, (8) where U = p q, A = A disp = ɛ /T ɛ γ/t ɛ /T ɛ, Q = α t + s() (p + γ)/t ɛ q/t ɛ. (9) The three real eigenvales of the Jacobian matri A disp can be shown to be (see also Fig. ) = L ɛ κ g T ɛ, = κ g, 3 = κ g, () and the corresponding right and left eigenvectors are 3 of 3

4 t t 3 3 O O (a) ɛ < (b) ɛ > Figre : Schematics of the dispersion waves strctres for the proposed hyperbolic dispersion system. Downloaded by NASA LANGLEY RESEARCH CENTRE on Agst 4, 6 DOI:.54/ R = κ g L ɛ L ɛ κ g L ɛ L ɛ κ g L ɛ L ɛ κ g, L = κ g + /L ɛ κ g/l ɛ κ g (κ g + )/L ɛ (κ g + )/L ɛ κ g + /L ɛ /(κ gl ɛ ) /κ g. () We now define the dispersion length scale, L ɛ, by first inserting the Forier mode of phase angle θ [, π] (i.e., U = U e Iθ/h ), where U is a constant vector, h is a grid spacing, and I =, into the hyperbolic dispersion system. We then evalate the eigenvales of the Forier-transformed operator with the κ g vale, and epand them for small θ to obtain the leading terms: λ = Iɛ h 3 θ3 + IɛL ɛ h 5 θ 5 + ɛl3 ɛ h 6 θ6 + O(θ 7 ), λ /3 = ɛ L 3 ɛ ± I ɛ L ɛh θ + O(θ ). () The first eigenvale is clearly an approimation to the Forier symbol of the operator ɛ, with a secondorder error in terms of θ. The other two represent damping modes with the real part, ɛ/l 3 ɛ, bt also have propagation in the net leading term, ±I ɛ θ/(l ɛh). To determine the dispersion length scale and possibly enhance the effects of error propagation dring transient psedo-time, we eqate the imaginary parts of the ɛ second and the third eigenvales with the leading term of the first eigenvale (i.e., h θ 3 = ɛ 3 L ɛh θ) for the smoothest mode (i.e., lowest freqency error mode) by setting θ = πh. After some algebra, we arrive at the following dispersion length scale: L ɛ = ± 4 π, (3) where the positive and negative length scales are associated with the positive and negative dispersion coefficient, ɛ, respectively. For all the nmerical eamples presented in Sec. VI, we observed eperimentally that both dobling and halving the above dispersion length scale reslt in a larger nmber of linear relaations and therefore, the above dispersion length scale appears to be optimm. The proposed hyperbolic dispersion system is now completely defined. In the net section, we present an etension of this system to an advective-dispersive PDE (sch as KdV) and show that a flly-hyperbolic system formlation of sch PDEs can also be fond. The general etension of the proposed hyperbolic system for a general advection-diffsion-dispersion eqation is discssed in Sec. IV. III. Hyperbolic Advection-Dispersion In this section, we briefly present an immediate etension of the hyperbolic dispersion system to general advective-dispersive PDEs (sch as KdV). Here, we present one possible flly-hyperbolic advective-dispersive system (in one-dimension), which cold be beneficial in imposing characteristic bondary conditions (BCs). The sbject of the characteristic BC is, however, beyond the scope of this paper and therefore, the effects of the formlation on characteristic BC is left for ftre investigations. Consider general advective-dispersive PDEs: t + f = ɛ, where advection speed is defined as 4 of 3

5 a() = f/. One possible hyperbolic formlation for advective-dispersive PDEs is: τ = a + ɛ q α t + s(), ( τ p = + ) ( p γ), T a T ɛ (4) τ q = T ɛ (γ + p q), Downloaded by NASA LANGLEY RESEARCH CENTRE on Agst 4, 6 DOI:.54/ where T a is the advection relaation time, γ = β + the following three characteristic speeds λ adv-disp = β ( T a + T ɛ ), λ adv-disp,3 = T ɛ ɛ T β a T ɛ ɛ β T, and T = +. This system has T a T ɛ ( a λ adv-disp ± ) (a λ adv-disp ) + 4ɛβ, (5) T ɛ with the corresponding right eigenvectors epressed as ɛ λ adv-disp T λ adv-disp 3 T R = ɛβ. (6) λ adv-disp a βt/t ɛ λ adv-disp /(ɛβ) βt/t ɛ λ adv-disp 3 /(ɛβ) We follow the same procedre as given in Sec. II, and obtain the relaation time as T = (alt ɛ a L T ɛ + 4LT ɛ (βl + ))/(ɛβ). With this relaation time, we can show that the eigenvales are always real, and matri of right eigenvectors is non-singlar with κ > (ecept for κ = ). Ths, the presented system is flly-hyperbolic for κ > and κ. An optimm length scale L can be obtained by a Forier analysis, as discssed in Sec. II. However, this comptation is not performed here, becase for general nmerical approimation we se the hyperbolic strctre of advection, diffsion, and dispersion operators separately (see Sec. IV). The individal treatment of these operators makes the etension to mlti-dimensions and/or more comple eqations straightforward. IV. Generalization: Hyperbolic Advection-Diffsion-Dispersion Here, we present an etension of the proposed hyperbolic dispersion formlation to a general hyperbolic advection-diffsion-dispersion system. This etension is not trivial and therefore, is reported here for completeness. Consider a nonlinear advection-diffsion-dispersion eqation t + (f) = (ν ) + ɛ, (7) where f is a nonlinear fnction of, and the diffsion coefficient, ν, may be a fnction of the soltion variable, and the advection speed is defined as a() = f/. Using the hyperbolic dispersion system introdced in Sec. II, we propose the following hyperbolic system for a general advection-diffsion-dispersion eqation, Eq. (7), τ = f + (νp + γν) + ɛ q α + s(), (8) ( t τ p = + ) ( p γ), (9) T ν T ɛ τ q = T ɛ (γ + p q), () which is formlated sch that it properly redces to a pre hyperbolic advection-dispersion and a pre hyperbolic advection-diffsion system, respectively, in the dispersion limit (ν ) and the diffsion limit (ɛ ). 5 of 3

6 Downloaded by NASA LANGLEY RESEARCH CENTRE on Agst 4, 6 DOI:.54/ Writing in vector form, Eq. (8), and following the non-nified approach of Ref. 3 for a separate treatment of advection, diffsion, and (in the this case) dispersion components, the fl Jacobian matri A of the proposed hyperbolic advection-diffsion-dispersion formlation becomes A = A adv + A diff + A disp = a γν + ν /T ν + A disp, () where A disp is given in Eq. (9). Note that for simplicity in the discssion, we presented the fl Jacobian for a linear ν, bt the proposed first-order system is also applicable for a nonlinear ν. Note also that we have inclded the fl Jacobian reslting from the presence of the γν term in the first eqation into the advective fl Jacobian, and not to the diffsion fl Jacobian; that is, we have treated γν as an added advection speed, which acts as an artificial advection in the diffsion limit (as γ does not vanish when ɛ ). With the above formlation, we also recover the identical hyperbolic diffsion system (which is in fact a [ ] system) of Ref. and ths, the same eigenvales, λ diff = ν/t ν, λ diff = ν/t ν, the diffsion relaation time, T ν = L ν/ν, and the diffsion length scale, L ν = /π, given in Ref.?, are applicable here. V. Discretization and Implicit Solver The proposed hyperbolic advection-diffsion-dispersion system can now be discretized with a desired scheme, sch as finite volme (FV), Discontinos-Galerkin (DG), RD, etc. Here we describe the discretization within the RD framework. Consider a one-dimensional domain discretized with N randomly distribted nodes. We store soltion vector U at each node denoted by j, where j =,, 3,..., N, and compte the cell residal, Φ E, by integrating the right-hand-side of Eq. (8) over the cell defined by the node j and j + (Fig. ): Φ E = j+ j ( A U + Q) d, () where we follow the techniqe of Ref. to compte cell residals of both the steady and nsteady sorce terms. j L E Figre : Schematic of the cell definition, and the left and the right nodes on each comptational cell. We now distribte the cell residal Φ E to the right and the left nodes of each cell with the following distribtion fnction, which is obtained sch that we can garantee conservation when advection, diffsion, and dispersion are treated separately: Φ L = B Φ E, Φ R = B + Φ E, B ± = I ± Dadv ± D diff ± D disp, (3) where the positive and the negative signs of the advection, D adv, the diffsion, D diff, and the dispersion, D disp, stabilization terms correspond to the distribtions of the cell residal Φ E to the right and the left nodes, respectively (see Fig. ). We define these stabilization terms independently by reformlating the advection, the diffsion, and the dispersion projection matrices in the form of the above distribtion fnction. To do this, we recall that an pwind distribtion matri can be constrcted as (after some algebra): B ± = R ± λ λ ± λ λ ± λ3 λ 3 R j L = I ± A ( R Λ L ) = I ± 3 A which reslts in the following advection, diffsion, and dispersion stabilization terms: D adv = Aadv /( a γν) + ɛ), D diff = Adiff l= λ diff l Πdiff l, D disp = Adisp l= l= 3 l λ l Π l, (4) Πdisp l, (5) 6 of 3

7 where ɛ is added to avoid division by zero when the total advection speed, a γν, is identically zero. Note that the distribtion matri sms p to the identity matri for each cell and therefore, we have conservation. The diffsion and dispersion projection matrices (i.e., Π diff and Π disp ), which are given here for convenience, are easily defined by projecting the diffsion and dispersion fles Jacobians onto their corresponding rnning waves (i.e., A diff. = l= λdiff l Π diff l, and A disp. = 3 l= λdisp l Π disp l ), to arrive at λ diff ν Π diff = /T ν, Π diff = I Π diff, (6) λ diff λ diff λ diff Downloaded by NASA LANGLEY RESEARCH CENTRE on Agst 4, 6 DOI:.54/ Π disp = κ g + κ g κ 3 gl ɛ κ gl ɛ κ g L ɛ κ 4 g κ 3 gl ɛ L ɛ κ g L ɛ κ g, Πdisp = L ɛ L ɛ L ɛ L ɛ L ɛ L ɛ, Π disp 3 = I Π disp Π disp, (7) where I is an identity matri. We now follow the procedre described in Ref., and constrct an implicit solver with a second-order compact Jacobian matri, which we compte nmerically with an atomatic-differentiation techniqe based on an operator-overloading algorithm. VI. Reslts In this section, we eamine and verify the accracy of the proposed hyperbolic advection-diffsiondispersion system. The presented eamples are solved by discretizing the hyperbolic system sing the RD scheme discssed in Sec. V. We rela the linearized system with a Gass-Seidel (GS) algorithm and redce the linear residals by three orders of magnitde with the maimm of relaations. This typically takes, on average, abot 4 5 GS relaations with an nder relaation parameter of.8. The implicit solver is then contined ntil ten orders of magnitde redction is achieved for all the eqations; this sally takes abot 4 5 Newton iterations (per time step). We first present an eample for solving a nonlinear advection-diffsion-dispersion eqation throgh the method-of-manfactred soltion. This eample is sed to verify the order of accracy of the scheme. We then present two more eamples; one for solving the classical KdV eqation with small dispersion coefficient, and one to demonstrate the capability of the proposed scheme in resolving physical oscillations in the zero dispersion limit. A. Eample : Order of accracy verification Consider the following nonlinear advection-diffsion-dispersion eqation: t + f = (ν ) + ɛ + s(, ) (8) where f = 3, ν =.5, ɛ = and s is the manfactred sorce term. We seek, throgh the method of manfactred soltion, a time-dependent two-soliton soltion of the (generalized) KdV eqation: 6 e (, t) = (η η ) ( η sech [χ(η )] + η csch [χ(η )] ) ( η tanh [χ(η )] η coth [χ(η )] ), (9) where χ(η) = η ( η t ã), (3) and η >, η >, and ã are arbitrary constants. Here we consider η =.5, η =., and ã =., bt similar reslts are also obtained with other vales. After reformlating Eq. (8) in the form of the proposed hyperbolic advection-diffsion-dispersion system discssed in Sec. IV, we apply the efficient forth-order RD scheme of Ref. for the spatial discretization, along with the A-stable second-order Backward-Differencing-Formla (BDF) for the temporal discretization 7 of 3

8 to the hyperbolic system. We then solve the system of eqations with an imposed Dirichlet bondary condition (i.e., the time-depended soltions at the bondary nodes are fied) and e (, ) as an initial condition. Figre 3 shows the predicted soltion and its first and second gradients (i.e., and ) at different times. These soltions are obtained with t =. sing 6 randomly distribted grid points in (, 3). As shown, the predicted forth-order reslts are in a very good agreement with the eact vales even on.5 (eact) (eact) (eact).5 (eact) (eact) (eact) Downloaded by NASA LANGLEY RESEARCH CENTRE on Agst 4, 6 DOI:.54/ (a) t = (c) t = 8. (eact) (eact) (eact) (b) t = 4. (eact) (eact) (eact) (d) t =. Figre 3: Two-soliton KdV with method of manfactred soltion: comparison with the predicted (forthorder spatial pls BDF) and eact soltions (inclding and ) at different times obtained on a randomly distribted nodes (N = 6) in [, 3] ( t =.). sch a relatively coarse grid. Figre 4 shows a comparison between the forth-order soltions and the second-order soltions for the same problem on the same randomly distribted grid. The presented reslts correspond to the soltion at t = with t =.. This figre shows that the second-order scheme over/nder predicts the peaks and valleys of the gradients, while the forth-order scheme captres them more accrately. We also verified the formal temporal and spatial orders of accracy for the soltion, gradient, and the Hessian (second-derivative) on a series of grids with randomly distribted nodes. The error convergence is 8 of 3

9 .5 Eact nd-order 4th-order.5 Eact nd-order 4th-order (,) (,) Downloaded by NASA LANGLEY RESEARCH CENTRE on Agst 4, 6 DOI:.54/ (a) at t = (b) at t = Figre 4: Comparison between the predicted second- and forth-order soltion gradients for the two-soliton KdV problem compted on a randomly distribted nodes (N = 6) in [, 3] (t =, t =.). obtained by compting the L norms of the soltions (i.e., L = N j= U j U e j /N, where U is the soltion vector, the sperscript e denotes the eact vale, and N is the total nmber of grid points). Similar reslts are also obtained with the L norm and therefore not shown. For temporal accracy verification, we sed a grid with 64 randomly distribted points, and varied t. For spatial accracy, we fied t =. and sed a series of randomly distribted grids. These reslts are shown in Fig. 5, indicating that all soltion variables, inclding the first two soltion gradients (i.e., and ) are forth-order accrate in space and second-order accrate in time. This is significant becase a sith-order accrate soltion is typically needed with conventional schemes to achieve forth-order accrate second-derivatives. The reconstrcted soltion gradients, which are compted sing a cbic polynomial fit of the forth-order compted soltion, are at least orders of magnitde less accrate than the compted soltion gradients. The orders of accracy of the reconstrcted soltion gradients are also shown in Fig. 5 for comparison. B. Eample : Colliding soliton waves Consider the following classical KdV eqation: t + f = ɛ. (3) where f = / and ɛ <. The following initial condition reslts in a time-dependent colliding two-soliton soltion: ( ) ( ) (, ) = 3 η sech η /ɛ [( ) η t] + 3 η sech η /ɛ [( ) η t] (3) where η and η are arbitrary positive constants, and and are the initial locations of the two soliton waves. Following the eample given in Ref. 7, we consider η =.3, η =., =.4, =.8 and ɛ = with a periodic bondary condition, and solve the classical KdV eqation in [, ] with randomly distribted nodes. Similar reslts are also obtained with other vales and therefore, not shown. Figre 6 shows the predicted soltion and its gradient and Hessian (i.e., and ) at t =,, 3. The initial soltion is also shown as reference. These soltions are obtained with t =. sing the forth-order (spatial) RD scheme (with BDF for time discretization) of Ref.. We note that these reslts are converged and frther grid refinement does not change the reslts qalitatively. 9 of 3

10 Hyperbolic Advection Diffsion Dispersion (Randomly Distribted Grid) Hyperbolic Advection Diffsion Dispersion (Randomly Distribted Grid) Slope Downloaded by NASA LANGLEY RESEARCH CENTRE on Agst 4, 6 DOI:.54/ log L Slope log t (a) Temporal accracy (BDF), N = 64 log L Slope 3 Slope 4 Slope log h (reconstrcted) (reconstrcted) (b) Spatial accracy (forth-order), t =. Figre 5: Temporal and spatial accracy verifications of the proposed hyperbolic advection-diffsiondispersion system for the soltion variable and its first two derivatives sing the forth-order RD-GT scheme (with BDF for temporal discretization) on randomly distribted nodes. C. Eample 3: Dispersive shock Consider a dispersive nonlinear Brger eqation (i.e., Eq. (3) when ɛ + ). This eqation has a dispersive behavior and prodces continos wavelets in the vicinity of the discontinity (aka dispersive shock waves). In this eample, we illstrate the capability of or code in resolving high-freqency wavelets for very small ɛ. Figre 7 shows the predicted reslts for ɛ = 3 and 4. The predicted high-order gradients and Hessians (second-derivatives) of the soltion are also provided. The reslts show that physical oscillations are captred and soltions are noise-free particlarly before and after the dispersive shocks (others reported soltions with some nmerical oscillations, see e.g., Ref. 4). For comparison, the reconstrcted gradient (third-order) and Hessian (second-order) are also shown for comparison. We note that these soltions are converged (i.e., no noticeable change in reslts are observed with frther grid refinement) and we eperienced no instability in obtaining these soltions. VII. Conclsions We have introdced, for the first time, a first-order system approach for general advection-diffsiondispersion eqations. We showed that the proposed system has real eigenvales with linearly independent eigenvectors, and ths is hyperbolic. We apply the forth-order RD scheme of Ref. to the proposed hyperbolic system, and solved several time-dependent dispersive eqations, inclding the classical two-soliton KdV and a dispersive shock case. We demonstrated that the predicted reslts, inclding the gradient and Hessian (i.e., and ), are in a very good agreement with the eact soltions. We also showed that the RD scheme applied to the proposed system can captre dispersive shocks with no nmerical oscillations. The design order of accracy of the scheme (forth-order spatial and second-order temporal) is also verified, and we achieved eqal order of accracy for the soltion, the gradient, and the Hessian. Acknowledgments The first and last athors wold like to thank the Center Chief Technology Office of NASA Langley Research Center for their spport throgh the Center Innovation Fnd (CIF). of 3

11 (a) t =, (b) t =, (c) t =, Downloaded by NASA LANGLEY RESEARCH CENTRE on Agst 4, 6 DOI:.54/ (d) t =, (g) t =, (e) t =, (h) t =, (f) t =, (i) t =, (j) t = 3, (k) t = 3, (l) t = 3, Figre 6: Two-soliton soltion to t + = ɛ : Predicted (forth-order spatial pls BDF) reslts with randomly distribted grid points in a periodic domain [, ] (ɛ = , t =.). of 3

12 Downloaded by NASA LANGLEY RESEARCH CENTRE on Agst 4, 6 DOI:.54/ (a) N = 3, t =.5, ɛ = (d) N = 5, t =.5, ɛ = 5 5 (reconstrcted) (b) N = 3, t =.5, ɛ = 4 5 (reconstrcted) (e) N = 5, t =.5, ɛ = 5 (reconstrcted) (c) N = 3, t =.5, ɛ = 4 5 (reconstrcted) (f) N = 5, t =.5, ɛ = 5 Figre 7: Dispersive shock case (zero dispersion limit): t + = ɛ, ɛ, (, ) = + / sin(π): predicted soltion, its gradient and Hessian (second-derivative) sing the forth-order (pls BDF) RD scheme applied to the proposed hyperbolic dispersion system on randomly distribted grid points in a periodic domain [, ]. of 3

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