Nonlinearly-Stable High-Order Methods on Simplices with Improved Efficiency

The summation by parts property is a discrete analogue of integration by parts. This property is fundamental to the weak formulation of problems, allowing for a scheme that simultaneously addresses both the strong and weak forms. This dual approach is particularly advantageous.

Traditionally, the summation by parts property has been associated with finite difference methods; however, its application extends to spectral element methods and discontinuous spectral elements. This property facilitates general proofs of stability by replicating the proof of well-posedness for continuous problems. While well-posedness for the Navier-Stokes equations remains unresolved, future research may provide insights that allow us to adapt these proofs.

Additionally, extensions to nonlinear problems can be achieved through entropy stability. We utilize simultaneous approximation terms for boundary conditions. Unlike conventional computational fluid dynamics (CFD) teaching, which often focuses on stability proofs in idealized scenarios, these stability proofs apply directly to the actual problems being solved, including situations involving distorted meshes.

To illustrate this concept, consider the linear convection equation in one dimension, defined over a specific domain with an initial condition and a boundary condition at the inflow. By multiplying the equation by the solution and integrating over the domain, we derive a relationship indicating that the total energy, represented by the integral of ( u^2 ) over the domain, is determined solely by the boundary conditions—specifically, what enters and exits the domain.

To ensure that our discrete formulation adheres to this principle, we introduce a derivative operator matrix. This matrix can be expressed as the product of the inverse of a symmetric positive definite norm matrix and another matrix ( Q ), which satisfies the required conditions in the simplest one-dimensional case.