Some people see the glass half full. Others see it half empty.

I see a glass that’s twice as big as it needs to be.

― George Carlin

Here, I will provide some concepts that may spur further development through attacking each of these three shortcomings in newer methods. Part of the route to analysis is utilization of evolution equations and the analysis of numerical methods using modified equation analysis. These concepts were essential in ideas that were key in the formulation of the original generation of (linear) monotone methods [HHL76] and revisiting these concepts should provide impetus for new advances. The approach is two-fold: explain why current methods are favored and provide a catalog of characteristics for new methods to follow.

In the past, I have used modified equation analysis in the past to unveil the form of the subgrid modeling provided by high-resolution methods for large eddy simulation (the so-called MILES or Implicit LES approach). This approach has provided a systematic approach to providing explanations for the effectiveness of high-resolution methods for modeling turbulence by revealing the implied subgrid model in implicit in the methods.

In solving hyperbolic conservation laws the work of Lax [Lax73] is preeminent. Lax identified the centrality of conservation form to computing weak solutions (along with Wendroff) [LW60]. Once weak solutions are achieved, it is essential to produce physically admissible solutions, which the concept of vanishing viscosity provides. Harten, Hyman and Lax [HHL76] provided the critical connection to numerical methods by showing that upwind differencing provided physically admissible solutions utilizing the method of modified equations to the analysis of the numerical method (the same can be shown for Lax-Friedrichs-based methods). These methods form the foundation upon which high-resolution methods are based.

Of course while these theoretical developments were made parallel discoveries were happening in the first high-resolution methods. Independently three different researchers discovered the basic principle of high-resolution methods, nonlinear selection of computational stencils so that the method becomes nonlinear even for linear problems [Boris71,VanLeer72,Kolgan72]. Each method allowed the barrier theorem of Godunov [God59] to be overcome and paved the way for further developments. Ultimately, Harten provided the work to tie these streams of work together by introducing the total variation concepts to formalize the mathematics [Harten83]. These methods form a foundation for a renaissance in numerical hyperbolic conservation laws.

Subsequently, there have been several attempts to move beyond the first generation of high resolution methods, but each has met with limited success [HEOC87,Shu87]. While this work has yielded some practical methods, none has shared the extreme success of the initial set of methods and their basis in mathematics. These methods have not yielded the same quantum leap in performance as the original methods, and not met the same success in being ubiquitous in applications. The reasons for this practical failure are multiple chiefly being cost, fragility, and failure of formal high-order accuracy to yield practical accuracy. I studied some of these basic tradeoffs with Jeff Greenough [GR04] between archetypical methods PLM [CG85] and fifth-order WENO [JS95].

The trick to forgetting the big picture is to look at everything close up.
― Chuck Palahniuk

I am proposing revisiting the foundational aspects of high-resolution methods by returning to modified equation analysis of numerical methods and providing connections to providing proof of physically admissible solutions, and their control of variation in the solution (i.e., connecting to variation-diminishing properties). To conduct such analysis I am suggesting the study of the energy evolution of equations, and/or the continuous variation evolution. Each equation will admit a vanishing viscosity solution, and the modified equation analysis can provide a connection of the numerical method to admissibility. The properties of the vanishing viscosity solution define the nature of desirable results. For a simple hyperbolic PDE, $\partial_t u + \partial_x f(u)=0$, the basic physical admissibility condition can be given $\partial_t u + \partial_x f(u)= \nu \partial_{xx} u, \nu\rightarrow 0$. For higher order methods, the viscosity must be nonlinear and a function of the solution itself. Similarly the variation evolution may be derived and its vanishing viscosity solution can be similarly stated. In the rest of the discussion here, we will focus on the linear version of the equation for simplicity, $\partial_t |V|+ \partial_u f \partial_x |V| = 0$.

When modified equation analysis is applied to upwind differencing, the classical result can be easily derived for the numerical viscosity, $\nu = \frac{1}{2}\Delta x |\partial_u f| + \text{HOT}$. For the variation equation the result is similar, but informative, $-|\partial_u f| |V|_{xx} -|\partial_u f| \partial_x\text{sign}(V) (V_x)^2$. We can see that the term proportional to $\partial_x\text{sign}(V)$ is positive negative and will reduce the variation where ever the sign of variation changes.

If we analyze classical TVD method using the minmod limiter, the connection to physical admissibility becomes clear for the kinetic energy evolution, $\nu =\frac{1}{4}\Delta x^2 |\partial_u f| |\frac{u_{xx}}{u_x}|+\text{HOT}$. Similarly the variation evolution provides a similar connection with more texture on the nature of the solution with the leading truncation error term being a flux $\text{sign}(V) \Delta x^3 \frac{1}{12}|\partial_u f| V_{xx} - \text{sign}(V) V_{xx} \frac{|V_x|V}{|V|V_x}$. When expanding the flux in the modified differential equation, two terms emerge as being non-conservative of variation, $(\Delta x)^2(\partial_u f\text{sign}(V) \frac{1}{12} V_x V_{xx} - \frac{1}{4} \partial_x \text{sign} (V_x) (V_{xx})^2)$. The second term is negative definite while the first term could be either positive or negative. We would categorize the scheme as being quite likely to be variation diminishing, but not absolutely. Discretely, we know that the method is TVD. Note that the viscosity coefficient, $|\frac{u_{xx}}{u_x}|$, is now a nonlinear function of the solution itself and could be called a “hyper-viscosity”. It is important that the actual discrete scheme limits the magnitude of this ratio to one, so the ratio does not blow up in practice.

We can apply the same technique to ENO or WENO methods and see the nature of its regularization at least for smooth solutions. For the famous fifth-order WENO we can derive the modified equation for the kinetic energy and find the leading order viscosity coefficient $|\partial_u f|(\Delta x)^5 \frac{(u_{xxx})^2}{20 u_x} - |\partial_u f|(\Delta x)^5 \frac{1}{60} u_{xxxxx}$, a nonlinear and linear hyper-viscous regularization. For the variation evolution the variation changing terms are $- |\partial_u f| (\Delta x)^5 \partial_x \text{sign}(V) \frac{1}{60} V_{xxxxxx} - |\partial_u f | (\Delta x)^5 \partial_x\text{sign}(V) \frac{(V_x)^2 (V_{xx})^2}{20 V^2} + |\partial_u f | (\Delta x)^5 \partial_x \text{sign}(V) \frac{V_x V_{xx} V_{xxx}}{10 |V|}$. The middle term is negative definite, but the first and third terms will be ambiguous for variation diminishment. This sort of behavior is observed for this scheme with the variation both shrinking and growing as time advances. Perhaps more importantly, the fully discrete method does not limit the size of the hyper-viscosity and it could blow up. The nonlinear hyperviscosity is itself ambiguous in terms of its variation diminishing character, which is distinctly different than the TVD scheme.

For the classical ENO scheme the result is quite similar. In the classical ENO scheme, the smoothest stencil is chosen hierarchically starting with first-order upwind. In other words the second order scheme is chosen to be the closest to the first-order scheme and the third-order stencil is chosen that is closest to the second-order stencil. We can analyze the third-order version using modified equation analysis. The effective numerical viscosity is a combination on linear and nonlinear hyperviscosity, $-\frac{1}{24}|\partial_u f|(\Delta x)^3 ( 4 |\frac{u_{xxx}}{u_{x}}| + |\frac{u_{xxx}}{u_{xx}}|\frac{u_{xx}}{u_x})$. This can be analyzed for the variation equation and produces ambiguously signed non-conservative terms, $-|\partial_u f|(\Delta x)^3 (\partial_x \text{sign}(V_{xx}) (V_{xxx})^3(\frac{1}{12}+ \text{sign}(V)\text{sign}(V_x)) + (\partial_x \text{sign}(V) V_x V_{xxx} (\frac{1}{24} + \frac{1}{24} \text{sign}(V_x)\text{sign}(V_{xx})))$. We find that ENO is quite similar to WENO, and the hyperviscosity magnitude is not controlled by the nonlinearity in the method. Practically speaking this is a deep concern when considering the robustness of the method.

The last method we analyze is one that I have suggested as a path forward. In a nutshell it is the high order method that is closest to a certain TVD method in some sense (I use the value of edge fluxes). Importantly, this method can still degenerate to the first-order upwind scheme while being formally third-order accurate. If the TVD method is chosen to be the first-order upwind method, we can analyze the result. The nonlinear dissipative flux is a third-order method, $|\partial_u f|(\Delta x)^3\frac{1}{24} u_{xxx} + |\partial_u f|(\Delta x)^3(\text{a big complicated mess})$. For the variation equation we get a remarkably simple non-conservative term that is negative definite, $- |\partial_u f||(\Delta x)^3\partial_x\text{sign} (V_{xx})\frac{(V_{xxx})^2}{12}$. This is an extremely hopeful form indicating excellent variation properties while retaining high order.

The modified equation analysis is enabled by a combination of symbolic algebra (Mathematica) and certain transformations formerly utilized to analyze implicit large eddy simulation [MR02,RM05,GMR07].

Finally, we can utilize these methods to produce a new class of methods that offer the promise of greater resolution and more provable connections to physical admissibility. The basic concept is to use TVD methods (and their close relatives) to ground high-order discretizations. The higher order approximations will be chosen to be closest to the TVD method is a well-defined sense. Through these means we can assure that the high-order method simultaneously provides accuracy and variation control by analyzing the resulting schemes via modified equation analysis. These methods will typically have similar regularizations to the WENO methods and provide high resolution results that are both formally accurate when appropriate and higher resolution than the TVD methods they are grounded by. For a third-order method the analysis of this method provides the following results for the kinetic energy equation, and the variation evolution. I believe this approach will provide a beneficial path for method development and analysis.

Out of clutter, find simplicity.
― Albert Einstein
If you’re not confused, you’re not paying attention.
― Tom Peters

[Harten83] Harten, Ami. “High resolution schemes for hyperbolic conservation laws.”Journal of computational physics 49, no. 3 (1983): 357-393.

[HEOC87] Harten, Ami, Bjorn Engquist, Stanley Osher, and Sukumar R. Chakravarthy. “Uniformly high order accurate essentially non-oscillatory schemes, III.” Journal of computational physics 71, no. 2 (1987): 231-303.

[HHL76] Harten, Amiram, James M. Hyman, Peter D. Lax, and Barbara Keyfitz. “On finite‐difference approximations and entropy conditions for shocks.”Communications on pure and applied mathematics 29, no. 3 (1976): 297-322.
[LW60] Lax, Peter, and Burton Wendroff. “Systems of conservation laws.”Communications on Pure and Applied mathematics 13, no. 2 (1960): 217-237.

[Lax73] Lax, Peter D. Hyperbolic systems of conservation laws and the mathematical theory of shock waves. Vol. 11. SIAM, 1973.

[Boris71] Boris, Jay P., and David L. Book. “Flux-corrected transport. I. SHASTA, A fluid transport algorithm that works.” Journal of computational physics 11, no. 1 (1973): 38-69.

(Boris, Jay P. A Fluid Transport Algorithm that Works. No. NRL-MR-2357. NAVAL RESEARCH LAB WASHINGTON DC, 1971.)

[VanLeer73] van Leer, Bram. “Towards the ultimate conservative difference scheme I. The quest of monotonicity.” In Proceedings of the Third International Conference on Numerical Methods in Fluid Mechanics, pp. 163-168. Springer Berlin Heidelberg, 1973.

[Kolgan72] Kolgan, V. P. “Application of the minimum-derivative principle in the construction of finite-difference schemes for numerical analysis of discontinuous solutions in gas dynamics.” Uchenye Zapiski TsaGI [Sci. Notes Central Inst. Aerodyn] 3, no. 6 (1972): 68-77.

(van Leer, Bram. “A historical oversight: Vladimir P. Kolgan and his high-resolution scheme.” Journal of Computational Physics 230, no. 7 (2011): 2378-2383.)

[God59] Godunov, Sergei Konstantinovich. “A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics.”Matematicheskii Sbornik 89, no. 3 (1959): 271-306.

[Shu87] Shu, Chi-Wang. “TVB uniformly high-order schemes for conservation laws.”Mathematics of Computation 49, no. 179 (1987): 105-121.

[GR04] Greenough, J. A., and W. J. Rider. “A quantitative comparison of numerical methods for the compressible Euler equations: fifth-order WENO and piecewise-linear Godunov.” Journal of Computational Physics 196.1 (2004): 259-281.

[CG85] Colella, Phillip, and Harland M. Glaz. “Efficient solution algorithms for the Riemann problem for real gases.” Journal of Computational Physics 59.2 (1985): 264-289. & Colella, Phillip. “A direct Eulerian MUSCL scheme for gas dynamics.” SIAM Journal on Scientific and Statistical Computing 6.1 (1985): 104-117.

[JS95] Jiang, Guang-Shan, and Chi-Wang Shu. “Efficient Implementation of Weighted ENO Schemes.” Journal of Computational Physics 1.126 (1996): 202-228.

[GLR07] Grinstein, Fernando F., Len G. Margolin, and William J. Rider, eds. Implicit large eddy simulation: computing turbulent fluid dynamics. Cambridge university press, 2007.

[RM05] Margolin, L. G., and W. J. Rider. “The design and construction of implicit LES models.” International journal for numerical methods in fluids 47, no. 10‐11 (2005): 1173-1179.

[MR02] Margolin, Len G., and William J. Rider. “A rationale for implicit turbulence modelling.” International Journal for Numerical Methods in Fluids 39, no. 9 (2002): 821-841.