Convex ENO Schemes for Hamilton-Jacobi Equations

Transcription

1 Convex ENO Schemes for Hamilton-Jacobi Equations Chi-Tien Lin Dedicated to our friend, Xu-Dong Liu, notre Xu-Dong. Abstract. In one dimension, viscosit solutions of Hamilton-Jacobi (HJ equations can be thought as primitives of entrop solutions for conservation laws. Based on this idea, both theoretical and numerical concepts used for conservation laws can be passed to HJ equations even in multi dimensions. In this paper, we construct convex ENO (CENO schemes for HJ equations. This construction is a generalization from the work b Xu-Dong Liu and S. Osher on CENO schemes for conservation laws. Several numerical experiments are performed. L and L error and convergence rate are calculated as well. Kewords. convex ENO schemes, Hamilton-Jacobi equations, conservation laws, two-dimensional Riemann problems Introduction We consider numerical solutions b convex ENO schemes (CENO for short for the Cauch problems of the Hamilton-Jacobi (HJ for short equation with Hamiltonian H: t ϕ + H( xϕ =, (. ϕ(x, = ϕ (x. HJ equations arise mainl from optimal control theor, differential games, geometric optics,... etc and are closel related to conservation laws (CL for short with fluxes f i, t u + n i= x i f i (u =, u(x, = u (x. (. Tpicall, solutions for HJ equations experience a loss of regularit as does the primitive of solutions to nonlinear CL. To be more precise, solutions of conservation laws in general, form discontinuities shocks, contact discontinuities et al., in a finite time even when the initial data are smooth. Indeed, for one-dimensional HJ equations ϕ is the viscosit solution of HJ equation (. with Hamiltonian H if and onl if u = x ϕ is the entrop solution of conservation laws (. with flux f(u = H( x ϕ and initial data u = ϕ []. In the multi-dimensional case, however, this kind of correspondence no longer exists. Instead, x ϕ satisfies a weakl hperbolic sstem of conservation laws [9]. In view of these arguments, we can think of viscosit solutions of the HJ equations (. as primitives of entrop solutions for the conservation laws (.. Based on this idea, both theoretical and numerical concepts used for conservation laws can be passed to HJ equations. (Consult, for example, [6] and the reference therein. In the last decade, several numerical schemes for conservation laws have been successfull extended to solve the HJ equations, for example, ENO schemes [], WENO schemes [8], relaxation schemes [9], discontinuous Galerkin methods[6], central schemes [5], central-upwind schemes []... etc. In this paper, we extend CENO schemes for conservation laws [7] to HJ equations. Department of Applied Mathematics, Providence Universit, Taichung 433, Taiwan. ctlin@pu.edu.tw

2 ENO schemes as developed in [5] and modified in [3] form a general method for solving sstems of hperbolic conservation laws in multi-dimensions. Modified ENO schemes use a conservation form approximation to point values, was implemented dimension b dimension, followed b an appropriate ODE solver, such as TVD Runge-Kutta methods. Modified ENO schemes have been implemented with great success since late 8 s. However, ENO-tpe schemes (and other upwind schemes evaluate their values over the same spatial points at all time steps. This in turn requires characteristic information along the discontinuous interface of these spatial points. Therefore, approximate Riemann solvers and, hence, field-b-field decomposition are needed to trace the characteristic fans. It is the field-b-field decomposition in which eigenvalues and both corresponding right and left eigenvectors of the Jacobian matrices are needed to be computed that makes ENO-tpe schemes expensive. On the other hand, central-tpe schemes compared to its counterpart, ENO-tpe schemes, are simplicit and efficienc. Because no need of field-b-field decomposition, central schemes can be implemented componentwise. However, the excessive numerical viscosit in the canonical first-order central difference scheme, the Lax-Friedrichs scheme, ields a relativel poor resolution, which seems to have delaed the development of a high-resolution central schemes. In [9,, 8, 7,,, 3] high order accurate sequels of centraltpe schemes have been designed, analzed and implemented with great success. Still, these methods were based on cell-average values and were, therefore, a bit complicated to extend to multi-dimensions. Consult, for example, []. The design of CENO schemes is to modif and extend this procedure of central schemes and ENO schemes so to obtain a third, or higher order component-wise central ENO scheme which is simple to implement in multi-dimensions. For ENO schemes, the interpolant is chosen to be the one which has the smaller (in magnitude derivative of possible candidates. The (ENO motivation for this choice was that choosing one or the other enables us to proceed to a higher degree polnomial, hence higher order accurate flux, in an hierarchical fashion. ENO schemes ma lead to oscillations when implemented in a component-wise wa. The main problem is that the numerical flux does not degenerate to the associated first order flux, eg, Lax-Friedrichs flux, at discontinuities. The ke feature of CENO schemes is then to introduce a new ENO-tpe decision to extend point (not cell-average values to obtain high order accurate numerical fluxes. Unlike ENO decision process, CENO decision process is designed to sta as close as possible to a flux which degenerates to first order at discontinuities, while maintaining higher order accurac at smooth region and which can be implemented componentwise. Based on the primitive relationship between solutions for HJ and for CL, we extend CENO schemes b Xu-Dong Liu and Osher [7] to HJ equations according to the conservative recipe introduced in []. This paper is organized as follows. In, we describe briefl the construction of third-order convex ENO schemes for conservation laws in [7]. We then construct CENO schemes for HJ equations in 3. In 4, numerical simulations are performed. Errors and convergence rates are calculated in both L - and L -frameworks. Acknowledgment: This was a joint work with Xu-Dong Liu. C.-T. Lin was partiall supported b NSC 9-5-M-6-3 and 93-5-M-6-. Part of the work was done while the author and Xu-Dong Liu visited the National Center for Theoretical Science, HsingChu, Taiwan, and the author visited Xu-Dong Liu at UCSB in June 4. Convex ENO schemes for conservation laws In this section, we describe the construction of a third-order convex ENO schemes for one dimensional hperbolic conservation laws in details. This construction can be generalized to multi-dimensional cases using dimension b dimension technique. We follow the original work b Liu and Osher [7]. Readers ma also consult [7] for more information.

3 We solve the Cauch problem of the one-dimensional conservation laws subject to initial data u (x: t u + xf(u =, (.3 u(x, = u (x. For simplicit, we assume a uniform computational grid: x j = j. We use u n j to denote the computed approximation to the exact solution u(x j, t n at the current time level, t = t n. ± p j = ±(p j± p j. We shall use conservative schemes of the full-discrete form with a consistent numerical flux u n+ j = u n j λ( ˆf j+ ˆf j, λ = t (.4 ˆf j+ = ˆf(u j l,..., u j+k, ˆf(u,..., u = f(u. (.5 Instead of using the full-discrete form (.5, we ma solve the conservation laws (.3 using a semi-discrete form d dt u j = L(u := ( ˆfj+ ˆf j (.6 together with an ODE solver for time discretization, for example a 3rd order TVD Runge-Kutta: q ( = q n + tl(q n q ( = 3 4 qn + 4 q( + 4 tl(q( q n+ = 3 qn + 3 q( + 3 tl(q(. Consult [], [4] and the reference therein for more high order ODE solvers. We start our construction of a 3rd order convex ENO scheme based on the first-order monotone Lax-Friedrichs (LF for short scheme. In general, f is not monotone. We first split flux b setting (.7 f + (u = (f(u + αu, f (u = (f(u αu (.8 where the parameter α max f (u so that f + (u and f (u. Then the LF flux at time level t n is f j+ = h +, j+/ + h, j+/ = [f(un j + f(u n j+] + α (un j + u n j+ (.9 where and h +, := f + (u j+ j = (f(u j + αu j, h, j+/ := f (u j+ = (f(u j+ αu j+. Algorithm:: (3rd order convex ENO scheme for CL Step : (Second-order interpolation of numerical fluxes The possible second-order numerical flux of f + (u at x j+ is γ +, := 3 f + (u j f + (u j over the stencil x j, x j } γ +, := f + (u j + f + (u j+ over the stencil x j, x j+ }. The possible second-order numerical flux of f (u at x j+ is γ, := f (u j + f (u j+ over the stencil x j, x j+ } γ, = 3 f (u j+ f (u j+ over the stencil x j+, x j+ }. 3

4 Step : (Convex ENO decision process For f +, if the st order numerical flux h +, j+ fluxes, γ +,, γ +,, then the nd order numerical flux h +, j+ lies between these two possible nd order numerical is the st order numerical flux h +, ; j+ otherwise, pick up the possible numerical flux which stas closest to the first-order flux as the nd order numerical flux. The resulted nd order numerical flux for f + is h +, j+ := h +, + ( j+ MM + h +,, h +, j+ j+ = f + (u n j + MM ( + f + j, f + j, Same process to choose the second-order numerical flux for f ields h, j+ Here MM is the minmod limiter: := h, ( j+ MM + h,, h, j+ j+ = f (u n j+ + MM ( + f j+, f j+. MM(x, = (sign x + sign min( x, (. Step 3: (3rd order interpolation of numerical fluxes The possible 3rd order numerical fluxes for f + at (x j+ are γ +,3 := 6 (f + 7f + + f + over x j 3 j j+ j, x j, x j } γ +,3 := 6 ( f + + 5f + + f + over x j j+ j+ 3 j, x j, x j+ } γ +,3 3 := 6 (f + + 5f + f + over x j+ j+ 3 j+ 5 j, x j+, x j+ }. The possible 3rd order numerical fluxes for f at (x j+ are γ,3 := 6 ( f + 5f + f over x j 3 j j+ j, x j, x j+ } γ,3 := 6 (f + 5f f over x j j+ j+ 3 j, x j+, x j+ } γ,3 3 := 6 (f 7f + f over x j+ j+ 3 j+ 5 j+, x j+, x j+3 }. Step 4: (convex ENO decision process For f +, if the second order numerical flux h +, j+ order numerical flux is h +,3 = h +, j+ j+ h +, j+. Same procedure to obtain h,3 j+ Step 5: (Time-marching Update u n+ j according to (.5 with flux ; otherwise h +,3 j+ for f from h, j+ ˆf LF,3 j+ lies among γ +,3, γ +,3 and γ +,3 3, then the third is one of γ +,3, γ +,3, γ +,3 3, which is closest to, γ,3, γ,3 and γ,3 3. = h +,3 + h,3. (. j+ j+ Before we turn to the construction of CENO schemes for HJ equations, a few remarks follow. Remark. Friedrich Instead of LF flux, we ma use other numerical fluxes, for example, the local Lax- f LLF j+ = f + (u j+ j + f (u j+ j+ 4

5 where with h + j+ = [f(u j + α j+ u j] and h = j+ [f(u j+ α j+ u j+] α j+ = max u I j f (u I j = (min(u j, u j+, max(u j, u j+. Note, that if f (u, on the interval above, then α j+ = max( f (u j, f (u j+. (. Remark. In above construction, we use minmod limiter for illustration. Indeed, it can be replaced b other limiters such as UNO limiter. Consult [7, ] and reference therein for other candidates. Remark.3 Due to numerical dissipation, numerical resolution b central schemes is lower compared to its counterpart, upwind schemes. To avoid the loss of accurac, one ma add in the CENO decision a free parameter α [] to select the most appropriate numerical flux from all possible ones. Remark.4 One can check easil that our second order numerical flux is the same for the second order non-oscillator TVD schemes. ˆf LF, j+ = f +, + f, j+ j+ = (f(u j+ + f(u j α(u j+ u j + 4 MM[ +f + (u j, f + (u j ] 4 MM[ +f (u j+, f (u j+ ] (.3 3 Convex ENO for H-J equations We now turn to the construction for a third order CENO scheme for HJ equation. For simplicit of exposition, we consider -dimensional HJ equations with Hamiltonian dependent on φ onl t ϕ + H(ϕ x, ϕ =, (3.4 ϕ(x,, = ϕ (x,. On the computational grid, x j := j and k := k, we use ϕ n jk, to denote the numerical approximation to the viscosit solution ϕ(x j, k, t n of the HJ equations (3.4 at the current time level t = t n. We also use the standard notations ± x ϕ n j,k = ±(ϕn j±,k ϕn j,k, and ± ϕ n j,k = ±(ϕn j,k± ϕn j,k. We shall take the full-discrete form ϕ n+ j,k = ϕn j,k tĥ(u+, u, v +, v (3.5 or the semi-discrete form dφ j = dt Ĥ(u+, u, v +, v (3.6 where the numerical Hamiltonian Ĥ is a Lipschitz continuous function consistent with H, that is Ĥ(u, u, v, v = H(u, v. A tpical example is the Lax-Freidrichs Hamiltonian: Ĥ LF (u +, u, v +, v = H( u+ + u where α x = max H u (u, v, α = max H v (u, v., v+ + v αx (u + u α (v + v We demonstrate our construction of a CENO scheme for the one-dimensional HJ equation based on the above Lax-Friedrichs Hamiltonian as follow 5

6 Algorithm:(3rd order CENO for HJ Step : (st order interpolation Set the st order interpolation to be φ +, j := + φ j and φ, j := φ j Step : (nd order interpolation The possible nd order approximation for φ +, j at x j are a +, = 3 + φ j + φ j+ a +, = + φ j + The possible nd order approximation for φ, j at x j are a, = 3 φ j φ j a, = φ j+ + over the stencil x j, x j+, x j+ } + φ j over the stencil x j, x j, x j+ }, over the stencil x j, x j, x j } φ j over the stencil x j, x j, x j+ }. Step 3: (convex ENO process If the st order approximation φ +, j lies between these two possible nd order approximation, a +,, a +,, then the nd order approximation φ +, j is the st order approximation φ +, j. Otherwise, pick up the possible approximation which stas closest to the first-order approximation as the nd order approximation. The resulted nd order approximation for u + is φ +, j = φ +, j + MM(a +, φ +, j, a +, φ +, j = + φ j φ j+ MM( + + φ j, + φ j Same process to choose the second-order approximation for u ields φ, j = φ, j + MM(a, φ, j, a, φ, j = φ j + φ j+ MM( φ j, φ j + φ j φ j Step 4: (3rd order interpolation Let S j be the stencil x j, x j, x j, x j+ }. The possible 3rd order approximation for φ +,3 j at x j are a +,3 = + φ j φ j 6 + φ j 6 over S j ; a +,3 = + φ j φ j φ j 6 over S j+ ; a +,3 = + φ j φ j φ j 6 over S j+. The possible 3rd order approximation for φ,3 j a,3 = φ j a,3 = 6 a,3 3 = 6 at x j are 6 7 φ j 6 + φ j 6 φ j+ + 5 φ j 6 φ j 6 φ j+ + 5 φ j over S j ; over S j ; φ j over S j+. Step 5: (convex ENO process If the second order approximation φ +, j lies among a +,3, a +,3 and a +,3 3, then the third order approximation is φ +,3 j = φ +, j ; otherwise φ +,3 j is one of a +,3, a +,3, a +,3 3, which is closest to φ +, j. The resulted formula is ( φ +,3 j = φ +, j + MM a +,3 φ +, j, a +,3 φ +, j, a +,3 3 φ +, j. Same procedure to obtain φ,3 j from φ, j, a,3, a,3 and a,3 φ,3 j = φ, j + MM 3. Resulted in ( a,3 φ, j, a,3 φ, j, a,3 3 φ, j. 6

7 Step 6: (Time marching Update φ n+ j according to (3.5, or (3.6 together with an appropriate ODE solver, with in the Hamiltonian ĤLF (u +, u. u + = φ +,3 j and u = φ,3 j (3.7 Remark 3. In above construction, we use the LF Hamiltonian: Ĥ LF (u +, u = H( u+ + u α(u+ u with α = max H. Indeed, we can choose other numerical Hamiltonian, for example Ĥ LLF (u +, u = H( u+ + u where I = [min(u, u +, max(u, u + ]. α(u+ u with α = max u I H (u Remark 3. For multi-dimensional HJ equations, we can compute CENO interpolation in each direction according to Steps -5 in the above Algorithm. Consult, for example, [] for details. Remark 3.3 With the LF Hamiltonian, our first order CENO for HJ reads φ n+ Ĥ = H( + φ j = H( + φ j+ φ j j = φ n j th( φn j+ φn j φ j φ j α( + φ j α (φ j+ + φ j α t (φn j+ + φn j and numerical Hamiltonian for the second order CENO schemes reads ( + φ j Ĥ = H + φ j φ j+ 4 MM( + + φ j, + φ j + φ j + φ j+ 4 MM( φ j, φ j φ j α [ + φ j φ j φ j+ MM( + + φ j, + φ j φ j+ MM( φ j, φ j φ j ] + φ j We note that the above formulae are slightl different to those central schemes for HJ equations introduced in [5]. 4 Numerical experiments In this section, we implement our 3rd order CENO schemes constructed in sections for both CL in 4. and HJ equations in 4.. 7

8 4. CENO for Conservation laws Numerical simulation on several tpical test problems of conservation laws using CENO was reported in the original work b X.-D. Liu and Osher [7]. We present here simulation on Riemann problems for two-dimensional Euler equations. Example 4. (Riemann Problems for Two-Dimensional Gas Dnamics where We consider Riemann problems for the two-dimensional Euler equations U = ρ ρu ρv e, F = U t + F (U x + G(U = (4.8 ρu ρu + p ρuv u(e + p, G = ρv ρuv ρv + p v(e + p (4.9 for a compressible gas and without the 4th component for an isentropic gas. Here ρ is the densit, u the velocit component in x-dimension, v the velocit component in -dimension, p the pressure, e the energ. p, ρ and e are connected b an equation of state that for an isentropic gas or a poltropic gas takes the form p = Aρ γ, e = p γ + ρ(u + v respectivel, here A > is a function of entrop. The Riemann problem is the initial value problem for (4.8 with initial data (p, ρ, u, v < x <, < < (p (p, ρ, u, v(x,, =, ρ, u, v < x <, < < (p 3, ρ 3, u 3, v 3 < x <, < < (p 4, ρ 4, u 4, v 4 < x <, < < for compressible gas. The total number of genuinel different configurations for poltropic gas is nineteen, see the following Table 4. We give out initial data for all configurations for a poltropic gas. Densit contour for each configuration is shown in Figures 4. and 4.. Our numerical results are strikingl consistent with calculations b Kurganov and Tadmor [3], and b Lax and Liu [4] for the same configurations with the same initial data. 8

9 Table 4.: Initial data for Riemann problems of D Euler equations. densit, x-velocit and -velocit respectivel. (pre, den, u, v are pressure, case time left right. up (,,,, (, 97, -59, down (.439,.7, -59, (, 79,, up (,,, (, 97, -59, down (,, -59, -59 (, 97,, up (.5,.5,, (, 33,.6, down (.9, 8,.6,.6 (, 33,, up (.,.,, (5, 65, 939, down (.,., 939, 939 (5, 65,, up (,, -5, - (,, -5, down (,, 5, (, 3, 5, up (,, 5, - (,, 5, down (,, -5, (, 3, -5, up (,,.,. (, 97, -59,. down (,,.,. (, 97,., up (, 97,.,. (,, -59,. down (,,.,. (,,., up (,,, (,,, - down (,.39,, -33 (, 97,, up (,,, 97 (,,, 76 down (333,,, -76 (333, 56,, up (,,., (, 33, 76, down (,,., (, 33,., 76.5 up (, 33,, (,, 76, down (,,, (,,, up (,,, - (,,, down (,.65,, 45 (, 33,, up (8,,, -66 (8,,, -.7 down (.6667, 736,,.7 (.6667, 474,, up (,,., - (, 97, -59, - down (,,., - (, 33,., up (, 33,.,. (,., -79,. down (,,.,. (,,, up (,,, - (,,, - down (,.65,, 5 (, 97,, up (,,, (,,, - down (,.65,, 5 (, 97,, up (,,, (,,, - down (,.65,, 5 (, 97,, -59 9

10 ..... (.. (... ( (4.. (5.. ( ( (7... ( (.. (... (..... (3 (6.. (4.. ( ( (8.. Figure 4.: D Riemenn problems via CENO with 4x4 points. Configuration -8.

11 .. (9.. Figure 4.: D Riemenn problems via CENO with 4x4 points. Configuration 9 No L norm L order L norm L order 4 5.7e e e-5 3..e e e e e e e-7 3. Table 4.: Error and order for the D Burgers equation 4. Hamilton-Jacobi equations Example 4. (Convergence rate test: one dimension We first solve the one-dimensional periodic HJ equations: ϕt + H(ϕ x =, x ϕ(x, = cos(πx, (4. with a strictl convex Hamiltonian H (Burgers-tpe equation: and a non-convex Hamiltonian H: H(u = (u + α. H(u = cos(u + α. The exact solution can be found through the solution of the corresponding conservation laws after changing variables consult Shu and Osher [] for details. For simplicit, we take α =. For the convex case, the singularit occurs at time t = /π, and near this time for the non-convex case. The third order convex ENO scheme with LF Hamiltonians is used for this example. We recorded data at t = /π before singularit exists. L - and L -errors and convergence rates are calculated and listed in Table 4. for Burgers equation and in Table 4.3 for the nonconvex HJ equation. We find that both L - and L -measure of the error in both cases achieves the expected convergence rate of order O( 3. The resolution of numerical solutions at t = /π (before the formation of singularit and at t =.5/π (after singularit for both equations are also presented in Figure 4.3. Example 4.3 (Two dimensional Burgers equations We solve the two-dimensional periodic HJ equations: ϕt + H(ϕ x, ϕ =, ϕ(x,, = cos(π x+, x, (4. with a strictl convex Hamiltonian H (Burgers-tpe equations: H(u, v = (u + v +

12 No L norm L order L norm L order 4.4e e e e e-6.95.e e e e-8.95.e-7.98 Table 4.3: Error and order for the D nonconvex HJ equation.... ( (..... (3 (4 Figure 4.3: Simulation for Example 4. with 4 points. (: Burgers equation, before singularit; (: Burgers equation, after singularit; (3: non-convex HJ equation, before singularit; (4: non-convex HJ equation, after singularit;

13 and a non-convex Hamiltonian H: H(u, v = cos(u + v +. Under the transformation, ξ = (x + and η = (x, our test problem (4. becomes the one-dimensional problem in the ξ direction, equation (4. in Example.[] We note that since we program in the (x, -coordinate sstem, this is a true two-dimensional problem. As in the Example 4., the singularit occurs at time t = /π for the convex Hamiltonian and near this time for the non-convex case. The LF Hamiltonian, are used in this example. Resolution results at time.5/π (after formation of singularit are shown in Figure (.5 x ( x Figure 4.4: Resolution after singularit, t =.5/π with 4x4 points. (,D Burgers equation; (,D non-convex equation Example 4.4 (Two dimensional kink. We solve another Cauch problem for a two-dimensional HJ equation with a non-convex Hamiltonian and a periodic boundar condition [9]: ϕt + ϕ x + ϕ + =, x,, (4. ϕ(x,, = (cos(πx (cos(π Using the 3rd order CENO scheme, we record data at t = (after singularit with mesh 5 5. the graph of the numerical solution is presented in Figure 4.5. The kink singularit has been carefull studied in [9] (. x.. ( x Figure 4.5: Resolution of ϕ for Example 4.4 via CENO 3rd-order with 5 5 points at t =. Example 4.5 (non-convex Riemann problem. We solve a two-dimensional non-convex Riemann problem with fixed boundar condition []. ϕt + sin(ϕ x + ϕ =, x,, ϕ(x,, = π( x (4.3 3

14 3.. ( 3 x ( x Figure 4.6: Resolution at t =. for Example 4.5 via the 3rd order CENO with 5 5 points We compute up to t =. with mesh 5 5. The numerical resolution is presented in Figure 4.6. Example 4.6 (Variable two-dimension. We solve the following problems related to control optimal cost determination: [, 8] ϕt sin(ϕ x + sin(xϕ + ϕ sin( + cos(x =, ϕ(x,, = (4.4 π x, π with periodic boundar condition. The Hamiltonian depends on not onl ϕ but also (x,. Numerical simulation b the second-order scheme, Algorithm 5, at t =. with mesh 5 5 is presented in Figure ( x 4 3 ( x 3 3 Figure 4.7: Resolution at t =. for Example 4.6 via 3rd order CENO with mesh: 5 5 References [] Corrias, L. and Falcone, M. and Natalini, R. (995 Numerical Schemes for Conservation Laws via Hamilton-Jacobi Eauations. Math. Comput. 64, [] Lev, D. and Puppo, G. and Russo, G. (999 Central WENO Schemes for Hperbolic Sstems of Conservation Laws. Math. Model. Numer. Anal. 33, [3] Lev, D. and Puppo, G. and Russo, G. ( Compact Central WENO Schemes for MultiDimensional Conservation Laws. SIAM J. Sci. Comput., [4] Gottlieb, S. and Shu, C.-W. and Tadmor, E. ( High order time discretization methods with the strong stabilit propert. SIAM Review 43, 89- [5] Harten, A and Engquist, B. and Osher, S. and Chakravarth, S. (987 Uniforml High Order Accurate Essentiall Non-Oscillator Schemes III. J. Comput. Phs. 7, 3-33,

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