improvement of accuracy and stability in numerically solving hyperbolic equations by ido...

Improvement of Accuracy and Stability in Numerically SolvingHyperbolic Equations by IDO (Interpolated DifferentialOperator) Scheme with Runge–Kutta Time Integration

Hiroshi Yoshida,1 Takayuki Aoki,2 and Takayuki Utsumi3

1Tokyo Institute of Technology, Tokyo, 152-8550 Japan

2Global Scientific Information and Computing Center, Tokyo Institute of Technology, Tokyo, 152-8550 Japan

3Advanced Photon Research Center Kansai Establishment, Japan Atomic Energy Research Institute, Kyoto, 619-0215 Japan

SUMMARY

In order to solve hyperbolic partial differential equa-tions by means of the Interpolated Differential Operator(IDO) scheme, time integration of the dependent variablehas been carried out by Taylor expansion, and time differ-entiation has been performed by replacing it with a spatialdifferentiation. However, in such a method, the time accu-racy is limited by the order of the interpolation function andin addition the spatial accuracy is not sufficient in multidi-mensional problems due to the complexity of the calcula-tions. In terms of numerical stability, the stable regionindicated by the CFL (Courant–Friedrich–Levy) number isnarrow. Hence, in order to improve the space–time accu-racy and to secure numerical stability, time integration bythe Runge–Kutta method is applied. Further, a method forincreasing the order of the Runge–Kutta method withoutincreasing the computational cost is proposed, taking ad-vantage of the characteristics of the IDO scheme with thephysical quantity and its spatial derivative as the dependentvariables. The two-dimensional advection equation and theone-dimensional wave equation are solved and the resultsare quantitatively compared with those obtained by theconventional Taylor expansion. Also, in order to demon-strate the adaptability of this approach to practical prob-lems, we consider Williamson’s test case 5 for theshallow-water equation in spherical geometry. It is found

that the Runge–Kutta method for multiple dimensionsyields accuracy and stability higher than those of the Taylorexpansion, demonstrating the effectiveness of the ap-proach. © 2003 Wiley Periodicals, Inc. Electron CommJpn Pt 3, 87(2): 33–42, 2004; Published online in WileyInterScience (www.interscience.wiley.com). DOI10.1002/ecjc.10127

Key words: IDO scheme; Runge–Kutta method;numerical stability; numerical accuracy; advection equa-tion.

1. Introduction

The authors have proposed the local InterpolatedDifferential Operator (IDO) scheme for accurate numericalcalculation of various partial differential equations [1–3].Briefly, the computational algorithm approximates the spa-tial profile of the physical quantity to be computed byhighly accurate interpolation functions and performshigher-order spatial integration and time differentiation bytreating the governing equation as the differential operatorfor the interpolation function. As the interpolation functionwe use Hermite interpolation, which is independently givenby the values of the physical quantity and the spatial deriva-

© 2003 Wiley Periodicals, Inc.

Electronics and Communications in Japan, Part 3, Vol. 87, No. 2, 2004Translated from Denshi Joho Tsushin Gakkai Ronbunshi, Vol. J86-A, No. 3, March 2003, pp. 223–231

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tive. The spatial derivative is determined by solving theequation derived by differentiating the governing equa-tion in a manner similar to the CIP method [4–7]. Sincethe interpolation form is usually constructed by using thevalue of the physical quantity and the first-order deriva-tive, the interpolation function between the two points iand i + 1 in one dimension is a third-order polynomial.When solving the advection equation, the time derivativeterm appearing in the Taylor expansion of the time canbe replaced by a spatial derivative by viewing the equa-tion as the differential operator. Since the order of theinterpolation function is up to third order even if theTaylor expansion is more than fourth order, it is foundthat the order of the interpolation function is the upperlimit of the accuracy of the numerical calculation. Whenmany mesh points are used for construction of a higher-order interpolation function, time integration with higheraccuracy becomes possible by carrying out Taylor expan-sion up to higher orders. However, the operation ofreplacing the time derivative with the spatial derivativebecomes more complex as the order becomes higher. Asthe equation becomes more complex, the computationalcost increases in accordance with a power series. Extensionto multiple dimensions becomes even more difficult and theaccuracy of the interpolation function may be lost. Also, itis a necessary condition in the explicit scheme that the CFL(Courant–Friedrich–Levy) number be less than unity fornumerical stability. In practice, an even smaller number isrequired for stability. If the Taylor expansion is used in timeintegration for multidimensional problems, the numericalaccuracy and stability are degraded. Even in the case of theCIP method, it is difficult to make calculations for morethan two dimensions with accuracy better than those forspatial third-order accuracy. Hence, in order to obtain com-putational accuracy comparable to that of the interpolationfunction regardless of the spatial dimensions, we proposeto carry out the time integration by the Runge–Kuttamethod instead of Taylor expansion.

In this paper, the second-, third-, and fourth-orderRunge–Kutta methods are used for the time integration ofthe IDO to solve a two-dimensional advection equation anda one-dimensional wave equation. The accuracy and stabil-ity are compared with those obtained by Taylor expansion.Further, a method is proposed to increase the numericalaccuracy of the Runge–Kutta method by one order withoutincreasing the computational cost by using the charac-teristics of the IDO. In order to confirm that this techniquecan be applied to practical problems, Williamson’s test case5 [8, 9] for the shallow-water equation in a spherical coor-dinate system is carried out.

2. Local Interpolated Differential Operator(IDO) Scheme

In the IDO scheme, the physical quantity and thespatial first order derivative are defined at each grid pointand the Hermite polynomial is locally formed with thesevalues (this polynomial then becomes an approximate so-lution). The unknown coefficients of the polynomial arederived from the matching conditions and the higher-orderspatial derivatives are derived by direct differentiation ofthe Hermite polynomial. The regions in which the interpo-lation functions are formed are different for the advectionterm and the nonadvection term. With regard to the advec-tion term, the upwind direction is detected from the advec-tion velocity at the grid point to be computed and aninterpolation function is formed for the upwind direction.This is called upwind interpolation. Both third-order up-wind interpolation using a third-order polynomial and fifth-order upwind interpolation using a fifth-order polynomialhave been described. They can be chosen depending on theaccuracy needed. Also, with regard to the nonadvectionterm, the interpolation function covers both the directionsat the center. This is called central interpolation, for whicha fifth-order polynomial is used.

2.1. Upwind interpolation

When third-order upwind interpolation is used, theHermite polynomial is expressed as F(x) =ax3 + bx2 + fx,ix + fi. Here, i is the grid index and fi and fx,idenote the physical quantity and its spatial first derivativeat grid point i. When the advection velocity is positive, aninterpolation function is formed from the (i – 1)-th gridpoint to the i-th grid point. Then, by solving the matchingconditions F(−∆x) = fi−1 and Fx(−∆x) = fx,i−1, the unknowncoefficients a and b are obtained as follows for determina-tion of the Hermite polynomial:

The higher-order spatial derivatives can be derived takingfxxx,i = Fxxx(0) = 6a and fxx,i = Fxx(0) = 2b. If the advectionvelocity is negative, the interpolation function is formedfrom the i-th to (i + 1)-th grid points and a similar processis followed.

When fifth-order upwind interpolation is issued, theHermite polynomial is F(x) = ax5 + bx4 + cx3 + dx2 +fx,ix + fi. If the advection velocity is positive, the interpola-tion function is formed from the (i – 2)-th grid point to the(i + 1)-th grid point. By solving the matching conditions

(1)

(2)

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F(−2∆x) = fi−2, F(− ∆x) = fi−1, F(∆x) = fi+1, and Fx(−∆x) =fx,i−1, the unknown coefficients a, b, c, and d are obtainedas follows for determination of the Hermite polynomial:

The higher-order spatial derivatives are fxxxxx = Fxxxxx(0) =120a, fxxxx = Fxxxx(0) = 24b, fxxx = Fxxx(0) = 6c, and fxx =Fxx(0) = 2d. If the advection velocity is negative, then theinterpolation function is formed from the (i – 1)-th gridpoint to the (i + 2)-th grid point and a similar process isfollowed.

2.2. Central interpolation

In fifth-order central interpolation, the Hermite poly-nomial can be expressed as F(x) = ax5 + bx4 + cx3 + dx2 +fx,ix + fi. The interpolation function is formed from the (i– 1)-th grid point to the (i + 1)-th grid point. By solvingF(−∆x) = fi−1, Fx(−∆x) = fx,i−1, F(∆x) = fi+1, and Fx(∆x) =fx,i+1, the unknown coefficients a, b, c, and d are obtainedas follows:

2.3. Time integration

In the conventional IDO scheme, the time integrationhas been carried in the IDO scheme by Taylor expansionwith respect to time. Since the physical quantity and itsspatial first derivative are in the dependent variable, timeintegration also must be carried out for these quantities:

In the above, the superscripts n and n + 1 indicate the valuesat times t and t + ∆t and ft indicates the derivative withrespect to time. The time derivatives are replaced with thespatial differentiations by means of the governing equation.For instance, for the one-dimensional advection equationft + ufx = 0 in a uniform velocity field, ftt = u2fxx, fttx =u2fxxx, fttt = u3fxxx, . . . so that all time derivatives are replacedwith the spatial derivatives.

3. Runge–Kutta Time Integration

We next present a method of applying the Runge–Kutta method to the time integration for the IDO scheme.In general, Runge–Kutta methods with accuracy of second,third, and fourth order are often used. In the following, anapplication with second order is explained. A similar pro-cedure can be applied to the third- and fourth-order cases.

3.1. Conventional method

When the one-dimensional governing equation hasthe form ft = ft(f, fx), the Runge–Kutta time integration ofthe dependent functions f and fx is as follows:

In the second-order Runge–Kutta method, the computationis up to K2. For the third and fourth orders, the computation

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is up to K3 and K4, respectively. The higher-order spatialderivatives appearing in Eq. (16) (such as fxx) are thoseobtained by differentiation of the interpolation functions.

If higher accuracy in time must be achieved in Taylorexpansion, it is necessary to derive higher-order time de-rivatives, for which the governing equation must be differ-entiated several more times. Hence, if the governingequation is complex, the computational cost is correspond-ingly larger.

On the other hand, in the Runge–Kutta method, thetime accuracy can be improved by iteratively using thegoverning equation (ft) and its spatial first derivative (ftx).Hence, no extra differentiation operations are needed andthe computational cost can be limited.

3.2. Improved method

Usually, in order to carry out time integration of n-thorder accuracy in the Runge–Kutta method, ft must becomputed more than n times. In this paper, by takingadvantage of the property that the physical quantity and itsspatial first derivative are dependent variables in the IDOscheme, we derive a method of obtaining (n + 1)-th orderaccuracy with n-th calculations. In the case of the second-order Runge–Kutta method, the following improvement isimplemented:

The key point is to replace K1 needed for derivationof K2 by the equation containing the second-order timederivative (ftt) of the physical quantity as shown in Eq. (21).Note that ftt can be derived readily from ft and ftx alreadyobtained. By this method, the accuracy of the calculationcan be increased by one order without significantly increas-ing the computational cost.

In the case of the third-order Runge–Kutta method,K1 is contained in the equations for K2 and K3 and hencethey should be replaced with equations containing second-order time derivatives of different physical quantities. Al-though this technique cannot be applied directly to thefourth-order Runge–Kutta method because K1 is containedonly in the equation for K2, it is possible to improve the

accuracy up to the fifth order by modifying the basicformula of the Runge–Kutta method. However, there areproblems of memory increase and cumbersome calcula-tions from the modification of the fourth-order Runge–Kutta method.

4. Numerical Experiments

The effectiveness of the Runge–Kutta method fortime integration in the IDO method is confirmed by com-parison with Taylor expansion. For check of the accuracyand numerical stability for the advection equation and forcomparison of the computation time, the two-dimensionaladvection equation in a uniform steady velocity field isused. For check of the accuracy for a nonadvection equa-tion, a one-dimensional wave equation is used.

4.1. Check of accuracy for advection equation

The two-dimensional advection equation in a uni-form steady velocity field is

In the IDO scheme, it is subsequently necessary to alsosolve ftx = −ufxx − vfxy, fty = −ufxy − vfyy, and ftxy = −ufxxy –vfxyy. Here, u and v are the advection velocities in the x andy directions. The calculation conditions are as follows. Thesteady velocity field is u = 1.0 and v = 1.0, the initial profileis f = 2 + sinx cosy, the computation domain is−π ≤ x ≤ π, −π ≤ y ≤ π, with the periodic boundary condi-tion. Let the CFL number (= ∆t/∆x) be 0.1 and let the errorbe evaluated from the numerical solution and the analyticalsolution at time 2π for the number of grid points (10 × 10),(20 × 20), . . . , (100 × 100). The equation used for evaluationis

where f and fanalytic indicate the numerical solution and theanalytic solution. Σallgrid implies summation over all gridpoints. Further, the fifth-order upwind interpolation is usedfor the advection term.

Figures 1 to 3 compare the numerical accuracy for theTaylor expansion and Runge–Kutta method when second-,third-, and fourth-order accuracy in time are taken. Even ifthe theoretical spatial accuracy is of fifth-order overall, it islimited by the time accuracy. Both agree up to the secondorder in time. The Runge–Kutta method provides betternumerical accuracy for the third and fourth orders. It isfound that even better numerical accuracy is possible, of the

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order of the time accuracy. On the other hand, in Taylorexpansion, the numerical accuracy stays at third order evenif the time accuracy is increased.

Figures 4 and 5 present the numerical accuracy ob-tained by the improved second- and third-order Runge–Kutta method. Although the accuracy is not equal to that inthe third- and fourth-order Runge–Kutta method, it is im-proved by one order.

4.2. Verification of accuracy of nonadvectionequation

The one-dimensional wave equation is given by

Fig. 1. The deviation from the exact solution versus thegrid interval in the two-dimensional advection equation.

A: Taylor expansion (2nd order); ": 2nd-orderRunge–Kutta scheme.


A: Taylor expansion (3rd order); ": 3rd-orderRunge–Kutta scheme.


A: Taylor expansion (4th order); ": 4th-orderRunge–Kutta scheme.


A: 2nd-order Runge–Kutta scheme; ": improved2nd-order Runge–Kutta scheme; $: 3rd-order

Runge–Kutta scheme.

(28)

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Since Eq. (28) contains second-order derivatives with re-spect to time, the following equations described in terms offirst-order time derivatives are used:

In the IDO scheme, it is also necessary to solve ftx = −gxx

and gtx = −fxx. The computation conditions are as follows.The initial profile is g = sinx and f = sinx, the computationaldomain is −π ≤ x ≤ π, with the periodic boundary condition.For grid numbers 10, 20, . . . , 100, the errors between thenumerical solutions and the analytical solutions at the time2π are evaluated by Eq. (27). It is assumed that CFL = ∆t/∆x= 0.2 from the propagation velocity of the wave (taken as 1here).

Figures 6 to 8 show a comparison of the numericalaccuracy of the Taylor expansion and the Runge–Kuttamethod for second-, third-, and fourth-order accuracy intime. They agree well for second- and third-order accuracyin time. When the accuracy is taken up to fourth order, theTaylor expansion provides more than fourth-order accuracybut the numerical accuracy of the Runge–Kutta method isfourth order.

Figures 9 and 10 study the numerical accuracy of theimproved second- and third-order Runge–Kutta methods.In each case, it is not possible to improve the numericalaccuracy by one order. Nevertheless, improvement is seen.

Also, it has been confirmed that fourth-order accu-racy is obtained for the two-dimensional wave equation.

4.3. Check of numerical stability

Check of numerical stability is performed by meas-uring the limit of the CFL numbers along each axis. TheCFL number in the steady velocity field is expressed by C= |u|∆t/∆x, where u is the velocity in the x direction. If the

(29)


A: 3rd-order Runge–Kutta scheme: ": improved3rd-order Runge–Kutta scheme; $: 4th-order

Fig. 6. The deviation from the exact solution versus thegrid interval in the one-dimensional sound wave

equation. A: Taylor expansion (2nd order); ": 2nd-order Runge–Kutta scheme.


equation. A: Taylor expansion (3rd order); ": 3rd-order Runge–Kutta scheme.

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CFL numbers in the x and y directions are CFLx and CFLy

and the grid distance and the time step interval are madeequal, then CFLx = |u| and CFLy = |v|. Here, u and v indicatethe velocities in the x and y directions. The computationconditions are as follows.

The number of grid points is 10 × 10, the initial profileis f = 2 + sinx cosy, and the computation continues up to thetime 200π. It is then concluded that the results are stable if

the value of Eq. (27) is less than 1.0. Under these conditions,u and v are varied independently from 0 to 1 and the stableregions of the Taylor expansion and Runge–Kutta methodsare investigated.

Figures 11 to 13 compare the results when accuracyof up to second, third, and fourth order in time is taken. Itis found that the shapes of the stable regions are completelydifferent for the two methods. When the second order intime is chosen, the stable region is wider in the Runge–Kutta method. It is evident from the figure that min(CFLx,


equation. A: Taylor expansion (4th order); ": 4th-order Runge–Kutta scheme.

Fig. 9. The deviation from the exact solution versus thegrid interval in the one-dimensional sound waveequation. A: 2nd-order Runge–Kutta scheme; ": improved 2nd-order Runge–Kutta scheme; $:

3rd-order Runge–Kutta scheme.

Fig. 10. The deviation from the exact solution versusthe grid interval in the one-dimensional sound wave

equation. A: 3rd-order Runge–Kutta scheme; ": improved 3rd-order Runge–Kutta scheme; $:

4th-order Runge–Kutta scheme.

Fig. 11. The stable domain in the 2nd-order accuracy inthe time step. ×: Runge–Kutta; +: Taylor expansion.

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CFLy) < 0.2 is the limit for stable calculations for Taylorexpansion. Here, min(A, B) implies choosing the smaller ofA and B. On the other hand, in the Runge–Kutta method,the limit is somewhat larger than that for Taylor expansion.Figures 14 and 15 show the stable regions for the improvedsecond- and third-order Runge–Kutta methods. They coin-cide with those before improvement and no increase ofstability is observed.

4.4. Comparison of computation time

The computation time was compared for each timeintegration scheme. The hardware and software used wereas follows:

Figure 16 shows the number of grid points in the x directionin each scheme (with an equal number of grid points in they direction) and the time needed for computation. TheRunge–Kutta method is faster for second-order accuracy intime while Taylor expansion is faster in the case of third-and fourth-order accuracy. The computational time for the

CPU Pentium-4, 1.3 GHz

Memory RDRAM 128 Mbyte

Compiler gcc version 2.95.2-13

Fig. 12. The stable domain in the 3rd-order accuracy inthe time step. ×: Runge–Kutta; +: Taylor expansion.

Fig. 13. The stable domain in the 4th-order accuracy inthe time step. ×: Runge–Kutta; +: Taylor expansion.

Fig. 14. The stable domain in the 2nd-orderRunge–Kutta scheme and improved

2nd-order Runge–Kutta scheme.

Fig. 15. The stable domain in the 3rd-orderRunge–Kutta scheme and improved

3rd-order Runge–Kutta scheme.

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improved Runge–Kutta method does not significantlychange from that before improvement. Hence, it is possibleto increase the numerical accuracy by the improved Runge–Kutta method without increasing the computational cost.

5. Application to Shallow-Water Equationsin Spherical Coordinates

As an instance of the application of time integrationby the Runge–Kutta method to practical problems, Wil-liamson’s test case 5 for the shallow-water equation in aspherical coordinate system was studied. The shallow-water equations consist of equations of conservation ofmomentum and conservation of mass. In this problem, ahigh mountain in middle latitudes is assumed and the timeevolution of the bypass flow is traced in this calculation.Figure 17 shows a contour map on the free surface after 15days. The results are almost identical to those obtained by

the pseudospectrum method. Hence, it is concluded that thescheme can be applied effectively to realistic simulations.

6. Conclusions

In this paper, the Runge–Kutta method is used as atime integration for the IDO scheme. Its effectiveness isconfirmed with regard to numerical accuracy, numericalstability, and computation time. The following conclusionsare obtained.

(1) In general, the overall computation accuracy is notimproved if the time accuracy is poor, even if the spatialaccuracy is high.

(2) In the case of one dimension, Taylor expansion issuperior in terms of numerical accuracy and numericalstability.

(3) In the case of two dimensions, the accuracy cannotbe improved up to the limit of the interpolation function byhigher-order expansion in Taylor expansion.

(4) In the case of two dimensions, the Runge–Kuttamethod has better numerical stability.

(5) It is possible to increase the numerical accuracyof the Runge–Kutta method by one order without increas-ing the computational cost by taking advantage of thecharacteristic of the IDO method that the physical quantityand its spatial first-order derivative are dependent variables.No improvement is seen in numerical stability.

(6) When the computation time is compared at thesame time accuracy, there is only a small difference be-tween Taylor expansion and the Runge–Kutta method.Hence, by using the improved Runge–Kutta scheme, thecomputation time can be made shorter than for Taylorexpansion.

(7) By solving Williamson’s test case 5 for the shal-low-water equation in spherical coordinates, it is found thatapplication to a realistic simulation is possible.

In computational fluid dynamics, high-accuracycomputation is needed for the advection term in the fluidequation. In order to make multidimensional fluid calcula-tions by the IDO scheme, it will be extremely useful to usethe Runge–Kutta method for the time integration.

REFERENCES

1. Aoki T. Interpolated Differential Operator (IDO)scheme for solving partial differential equations.Comput Phys Commun 1997;102:132–146.

2. Sakurai K, Aoki T. Implicit IDO (Interpolated Differ-ential Operator) scheme. Comput Fluid Dynamics J1999;8:6–12.

Fig. 16. Comparison of time performance in each timeadvance scheme in the two-dimensional advection

equation.

Fig. 17. Application to Williamson’s test case 5.Contour map on the free surface after 15 days.

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3. Aoki T. 3D simulation for falling papers. ComputPhys Commun 2001;142:326–329.

4. Yabe T, Aoki T. A universal solver for hyperbolicequations by cubic-polynomial interpolation I. Onedimensional solver. Comput Phys Commun1991;66:219–232.

5. Aoki T. Multi-dimensional advection of CIP (Cubic-Interpolated Propagation) scheme. Comput FluidDynamics J 1995;4:279–291.

6. Yabe T, Xiao F, Utsumi T. The constrained interpola-tion profile method for multiphase analysis. J Com-put Phys 2001;169:556–593.

7. Utsumi T. Differential algebraic hydrodynamicssolver with cubic-polynomial interpolation. CFD J1995;4:225–238.

8. Williamson DL, Drake JB, Hanck JJ, Jakob R,Swarztrauber PN. A standard test set for numericalapproximations to the shallow water equations inspherical geometry. J Comput Phys 1992;102:211–224.

9. Jakob-Chen R, Hanck JJ, Williamson DL. Spectraltransform solutions to the shallow water test set. JComput Phys 1995;119:164–187.

AUTHORS (from left to right)

Hiroshi Yoshida (student member) graduated from the Department of Applied Physics, Tokyo Institute of Technology,in 2000 and began graduate studies. He has been engaged in ocean simulations.

Takayuki Aoki (member) graduated from the Department of Applied Physics, Tokyo Institute of Technology, in 1983,completed the M.S. program in 1985, and joined the Fujitsu Atsugi Research Laboratory. In 1986, he became a research associateat Tokyo Institute of Technology, and has been a professor since 2001. His research areas are numerical fluid mechanics andlarge-scale simulations. He is a member of the Japan Mechanical Engineering Society and the Information Processing Society.

Takayuki Utsumi graduated from the Department of Physics, Tokyo Institute of Technology, in 1974 and joined JapanAviotronics. In 1976, he moved to Mitsubishi Space Software Co., Ltd. He has been with the Japan Atomic Energy ResearchInstitute since 1996, and is engaged in research on photon simulation.

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improvement of accuracy and stability in numerically solving hyperbolic equations by ido...

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