Godunov's theorem

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 methods 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 ${\displaystyle x_{j}=j\,\Delta x}$  and ${\displaystyle t^{n}=n\,\Delta t}$ , such a scheme can be described by

${\displaystyle \sum \limits _{m=1}^{M}{\beta _{m}}\varphi _{j+m}^{n+1}=\sum \limits _{m=1}^{M}{\alpha _{m}\varphi _{j+m}^{n}}.\quad \quad (1)}$ 

In other words, the solution ${\displaystyle \varphi _{j}^{n+1}}$  at time ${\displaystyle n+1}$  and location ${\displaystyle j}$  is a linear function of the solution at the previous time step ${\displaystyle n}$ . We assume that ${\displaystyle \beta _{m}}$  determines ${\displaystyle \varphi _{j}^{n+1}}$  uniquely. Now, since the above equation represents a linear relationship between ${\displaystyle \varphi _{j}^{n}}$  and ${\displaystyle \varphi _{j}^{n+1}}$  we can perform a linear transformation to obtain the following equivalent form,

${\displaystyle \varphi _{j}^{n+1}=\sum \limits _{m}^{M}{\gamma _{m}\varphi _{j+m}^{n}}.\quad \quad (2)}$ 

Theorem 1: Monotonicity preserving

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

${\displaystyle \gamma _{m}\geq 0,\quad \forall m.\quad \quad (3)}$ 

Proof - Godunov (1959)

Case 1: (sufficient condition)

Assume (3) applies and that ${\displaystyle \varphi _{j}^{n}}$  is monotonically increasing with ${\displaystyle j}$ .

Then, because ${\displaystyle \varphi _{j}^{n}\leq \varphi _{j+1}^{n}\leq \cdots \leq \varphi _{j+m}^{n}}$  it therefore follows that ${\displaystyle \varphi _{j}^{n+1}\leq \varphi _{j+1}^{n+1}\leq \cdots \leq \varphi _{j+m}^{n+1}}$  because

${\displaystyle \varphi _{j}^{n+1}-\varphi _{j-1}^{n+1}=\sum \limits _{m}^{M}{\gamma _{m}\left({\varphi _{j+m}^{n}-\varphi _{j+m-1}^{n}}\right)}\geq 0.\quad \quad (4)}$ 

This means that monotonicity is preserved for this case.

Case 2: (necessary condition)

We prove the necessary condition by contradiction. Assume that ${\displaystyle \gamma _{p}^{}<0}$  for some ${\displaystyle p}$  and choose the following monotonically increasing ${\displaystyle \varphi _{j}^{n}\quad }$ ,

${\displaystyle \varphi _{i}^{n}=0,\quad i<k;\quad \varphi _{i}^{n}=1,\quad i\geq k.\quad \quad (5)}$ 

Then from equation (2) we get

${\displaystyle \varphi _{j}^{n+1}-\varphi _{j-1}^{n+1}=\sum \limits _{m}^{M}{\gamma _{m}}\left({\varphi _{j+m}^{n}-\varphi _{j+m-1}^{n}}\right)=\left\{{\begin{array}{*{20}c}{0,}&{\left[{j+m\neq k}\right]}\\{\gamma _{m},}&{\left[{j+m=k}\right]}\\\end{array}}\right.\quad \quad (6)}$ 

Now choose ${\displaystyle j=k-p}$ , to give

${\displaystyle \varphi _{k-p}^{n+1}-\varphi _{k-p-1}^{n+1}={\gamma _{p}\left({\varphi _{k}^{n}-\varphi _{k-1}^{n}}\right)}<0,\quad \quad (7)}$ 

which implies that ${\displaystyle \varphi _{j}^{n+1}}$  is NOT increasing, and we have a contradiction. Thus, monotonicity is NOT preserved for ${\displaystyle \gamma _{p}<0}$ , which completes the proof.

Theorem 2: Godunov’s Order Barrier Theorem

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

${\displaystyle {{\partial \varphi } \over {\partial t}}+c{{\partial \varphi } \over {\partial x}}=0,\quad t>0,\quad x\in \mathbb {R} \quad \quad (10)}$ 

cannot be monotonicity preserving unless

${\displaystyle \sigma =\left|c\right|{{\Delta t} \over {\Delta x}}\in \mathbb {N} ,\quad \quad (11)}$ 

where ${\displaystyle \sigma }$  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

${\displaystyle \varphi \left({0,x}\right)=\left({{x \over {\Delta x}}-{1 \over 2}}\right)^{2}-{1 \over 4},\quad \varphi _{j}^{0}=\left({j-{1 \over 2}}\right)^{2}-{1 \over 4}.\quad \quad (12)}$ 

The exact solution is

${\displaystyle \varphi \left({t,x}\right)=\left({{{x-ct} \over {\Delta x}}-{1 \over 2}}\right)^{2}-{1 \over 4}.\quad \quad (13)}$ 

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

${\displaystyle \varphi _{j}^{1}=\left({j-\sigma -{1 \over 2}}\right)^{2}-{1 \over 4},\quad \varphi _{j}^{0}=\left({j-{1 \over 2}}\right)^{2}-{1 \over 4}.\quad \quad (14)}$ 

Substituting into equation (2) gives:

${\displaystyle \left({j-\sigma -{1 \over 2}}\right)^{2}-{1 \over 4}=\sum \limits _{m}^{M}{\gamma _{m}\left\{{\left({j+m-{1 \over 2}}\right)^{2}-{1 \over 4}}\right\}}.\quad \quad (15)}$ 

Suppose that the scheme IS monotonicity preserving, then according to the theorem 1 above, ${\displaystyle \gamma _{m}\geq 0}$ .

Now, it is clear from equation (15) that

${\displaystyle \left({j-\sigma -{1 \over 2}}\right)^{2}-{1 \over 4}\geq 0,\quad \forall j.\quad \quad (16)}$ 

Assume ${\displaystyle \sigma >0,\quad \sigma \notin \mathbb {N} }$  and choose ${\displaystyle j}$  such that ${\displaystyle j>\sigma >\left(j-1\right)}$  . This implies that ${\displaystyle \left({j-\sigma }\right)>0}$  and ${\displaystyle \left({j-\sigma -1}\right)<0}$  .

It therefore follows that,

${\displaystyle \left({j-\sigma -{1 \over 2}}\right)^{2}-{1 \over 4}=\left(j-\sigma \right)\left(j-\sigma -1\right)<0,\quad \quad (17)}$ 

which contradicts equation (16) and completes the proof.

The exceptional situation whereby ${\displaystyle \sigma =\left|c\right|{{\Delta t} \over {\Delta x}}\in \mathbb {N} }$  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

• Finite volume method
• Flux limiter
• Total variation diminishing

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, Mat. 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.