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In numerical analysis and computational fluid dynamics, Godunov's theorem — also known as Godunov's order barrier theorem — is a mathematical theorem important in the development of the theory of high resolution schemes for the numerical solution of partial differential equations.

The theorem states that:

Linear numerical schemes for solving partial differential equations (PDE's), having the property of not generating new extrema (monotone scheme), can be at most first-order accurate.

Professor Sergei K. Godunov originally proved the theorem as a Ph.D. student at Moscow State University. It is his most influential work in the area of applied and numerical mathematics and has had a major impact on science and engineering, particularly in the development of methodologies used in computational fluid dynamics (CFD) and other computational fields. One of his major contributions was to prove the theorem (Godunov, 1954; Godunov, 1959), that bears his name.

The theorem

We generally follow Wesseling (2001).

Aside

Assume a continuum problem described by a PDE is to be computed using a numerical scheme based upon a uniform computational grid and a one-step, constant step-size, M grid point, integration algorithm, either implicit or explicit. Then if xj=jΔx and tn=nΔt, such a scheme can be described by

m=1Mβmφj+mn+1=m=1Mαmφj+mn.(1)

In other words, the solution φjn+1 at time n+1 and location j is a linear function of the solution at the previous time step n. We assume that βm determines φjn+1 uniquely. Now, since the above equation represents a linear relationship between φjn and φjn+1 we can perform a linear transformation to obtain the following equivalent form,

φjn+1=mMγmφj+mn.(2)

Theorem 1: Monotonicity preserving

The above scheme of equation (2) is monotonicity preserving if and only if

γm0,m.(3)

Proof - Godunov (1959)

Case 1: (sufficient condition)

Assume (3) applies and that φjn is monotonically increasing with j.

Then, because φjnφj+1nφj+mn it therefore follows that φjn+1φj+1n+1φj+mn+1 because

φjn+1φj1n+1=mMγm(φj+mnφj+m1n)0.(4)

This means that monotonicity is preserved for this case.

Case 2: (necessary condition)

We prove the necessary condition by contradiction. Assume that γp<0 for some p and choose the following monotonically increasing φjn,

φin=0,i<k;φin=1,ik.(5)

Then from equation (2) we get

φjn+1φj1n+1=mMγm(φj+mnφj+m1n)={0,[j+mk]γm,[j+m=k](6)

Now choose j=kp, to give


φkpn+1φkp1n+1=γp(φknφk1n)<0,(7)

which implies that φjn+1 is NOT increasing, and we have a contradiction. Thus, monotonicity is NOT preserved for γp<0, which completes the proof.

Theorem 2: Godunov’s Order Barrier Theorem

Linear one-step second-order accurate numerical schemes for the convection equation

φt+cφx=0,t>0,x(10)

cannot be monotonicity preserving unless

σ=|c|ΔtΔx,(11)

where σ is the signed Courant–Friedrichs–Lewy condition (CFL) number.

Proof - Godunov (1959)

Assume a numerical scheme of the form described by equation (2) and choose

φ(0,x)=(xΔx12)214,φj0=(j12)214.(12)

The exact solution is

φ(t,x)=(xctΔx12)214.(13)

If we assume the scheme to be at least second-order accurate, it should produce the following solution exactly

φj1=(jσ12)214,φj0=(j12)214.(14)

Substituting into equation (2) gives:

(jσ12)214=mMγm{(j+m12)214}.(15)

Suppose that the scheme IS monotonicity preserving, then according to the theorem 1 above, γm0.

Now, it is clear from equation (15) that

(jσ12)2140,j.(16)

Assume σ>0,σ and choose j such that j>σ>(j1) . This implies that (jσ)>0 and (jσ1)<0 .

It therefore follows that,

(jσ12)214=(jσ)(jσ1)<0,(17)

which contradicts equation (16) and completes the proof.

The exceptional situation whereby σ=|c|ΔtΔx is only of theoretical interest, since this cannot be realised with variable coefficients. Also, integer CFL numbers greater than unity would not be feasible for practical problems.

See also

References

  • Godunov, Sergei K. (1954), Ph.D. Dissertation: Different Methods for Shock Waves, Moscow State University.
  • Godunov, Sergei K. (1959), A Difference Scheme for Numerical Solution of Discontinuous Solution of Hydrodynamic Equations, Math. Sbornik, 47, 271-306, translated US Joint Publ. Res. Service, JPRS 7226, 1969.
  • Wesseling, Pieter (2001), Principles of Computational Fluid Dynamics, Springer-Verlag.

Further reading

  • Hirsch, C. (1990), Numerical Computation of Internal and External Flows, vol 2, Wiley.
  • Laney, Culbert B. (1998), Computational Gas Dynamics, Cambridge University Press.
  • Toro, E. F. (1999), Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer-Verlag.
  • Tannehill, John C., et al., (1997), Computational Fluid mechanics and Heat Transfer, 2nd Ed., Taylor and Francis.