John N. Shadid

Sidafa Conde, Sigal Gottlieb, Zachary J. Grant et al.

When evolving in time the solution of a hyperbolic partial differential equation, it is often desirable to use high order strong stability preserving (SSP) time discretizations. These time discretizations preserve the monotonicity properties satisfied by the spatial discretization when coupled with the first order forward Euler, under a certain time-step restriction. While the allowable time-step depends on both the spatial and temporal discretizations, the contribution of the temporal discretization can be isolated by taking the ratio of the allowable time-step of the high order method to the forward Euler time-step. This ratio is called the strong stability coefficient. The search for high order strong stability time-stepping methods with high order and large allowable time-step had been an active area of research. It is known that implicit SSP Runge-Kutta methods exist only up to sixth order. However, if we restrict ourselves to solving only linear autonomous problems, the order conditions simplify and we can find implicit SSP Runge-Kutta methods of any linear order. In the current work we aim to find very high linear order implicit SSP Runge-Kutta methods that are optimal in terms of allowable time-step. Next, we formulate an optimization problem for implicit-explicit (IMEX) SSP Runge-Kutta methods and find implicit methods with large linear stability regions that pair with known explicit SSP Runge-Kutta methods of orders plin=3,4,6 as well as optimized IMEX SSP Runge-Kutta pairs that have high linear order and nonlinear orders p=2,3,4. These methods are then tested on sample problems to verify order of convergence and to demonstrate the sharpness of the SSP coefficient and the typical behavior of these methods on test problems.

Sidafa Conde, Imre Fekete, John N. Shadid

We construct a family of embedded pairs for optimal strong stability preserving explicit Runge-Kutta methods of order $2 \leq p \leq 4$ to be used to obtain numerical solution of spatially discretized hyperbolic PDEs. In this construction, the goals include non-defective methods, large region of absolute stability, and optimal error measurement as defined in [5,19]. The new family of embedded pairs offer the ability for strong stability preserving (SSP) methods to adapt by varying the step-size based on the local error estimation while maintaining their inherent nonlinear stability properties. Through several numerical experiments, we assess the overall effectiveness in terms of precision versus work while also taking into consideration accuracy and stability.

Jeonghun J. Lee, Stephen Shannon, Tan Bui-Thanh et al.

We present a hybridized discontinuous Galerkin (HDG) method for stationary linearized incompressible magnetohydrodynamics (MHD) equations. At the heart of the paper is the introduction of an HDG flux of the dual saddle-point form of the MHD equations that facilitates the hybridization of discontinuous Galerkin (DG) method. We carry out the $\textit{a priori}$ error estimates for the proposed HDG method on simplicial meshes in both two- and three-dimensions. The analysis provides optimal convergence for the fluid velocity and the magnetic variables, and quasi-optimal convergence for the remaining quantities. Numerical examples are presented to verify the theoretical findings.

Jehanzeb H. Chaudhry, J. B. Collins, John N. Shadid

Implicit-Explicit (IMEX) schemes are widely used for time integration methods for approximating solutions to a large class of problems. In this work, we develop accurate a posteriori error estimates of a quantity of interest for approximations obtained from multi-stage IMEX schemes. This is done by first defining a finite element method that is nodally equivalent to an IMEX scheme, then using typical methods for adjoint-based error estimation. The use of a nodally equivalent finite element method allows a decomposition of the error into multiple components, each describing the effect of a different portion of the method on the total error in a quantity of interest.