Optimal Explicit Strong Stability Preserving Runge--Kutta Methods with High Linear Order and optimal Nonlinear Order

High order spatial discretizations with monotonicity properties are often desirable for the solution of hyperbolic PDEs. These methods can advantageously be coupled with high order strong stability preserving time discretizations. The search for high order strong stability time-stepping methods with large allowable strong stability coefficient has been an active area of research over the last two decades. This research has shown that explicit SSP Runge--Kutta methods exist only up to fourth order. However, if we restrict ourselves to solving only linear autonomous problems, the order conditions simplify and this order barrier is lifted: explicit SSP Runge--Kutta methods of any linear order exist. These methods reduce to second order when applied to nonlinear problems. In the current work we aim to find explicit SSP Runge--Kutta methods with large allowable time-step, that feature high linear order and simultaneously have the optimal fourth order nonlinear order. These methods have strong stability coefficients that approach those of the linear methods as the number of stages and the linear order is increased. This work shows that when a high linear order method is desired, it may be still be worthwhile to use methods with higher nonlinear order

Implicit and Implicit-Explicit Strong Stability Preserving Runge-Kutta Methods with High Linear Order

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

Meta-analysis of repair techniques for mixed nerve damage

BACKGROUND: Injuries to the upper extremity are common and can occur with trauma, surgery, or compression. Particularly, injuries that result in complete transection often require surgical intervention in attempt to restore function and quality of life. Classically, nerve autografts have been used as the gold standard to repair these peripheral lesions. An alternative to the autologous nerve graft is the use of the processed nerve allograft. limited research exists in comparing sensory and motor outcomes of allograft nerve repair in the upper extremity with data for autograft repair. In this systematic review and meta-analysis, we set out to cumulate results of motor and sensory outcomes of allograft peripheral nerve lesion repair and compare those results with outcomes of autograft repair. METHODS: Current literature on motor and sensory outcomes of autograft and allograft peripheral nerve repair were reviewed using British Medical Research Council (MRC) score of sensory recovery or British Medical Research Council (MRC) muscle strength grading system and complication rates as outcomes of interest. After inclusion and exclusion criteria were applied, 12 articles were reviewed and 826 nerve repairs were analyzed. RESULTS: The mean gap length for the allograft group and autograft group was 28.6 mm and 24.7 mm, respectively. In terms of MRC sensory and motor outcomes, allograft repair was statistically superior to autograft repair. Complication data was insignificant. CONCLUSIONS: Based on the current updated meta-analysis using recent data, we found that both autograft and allograft repair have reasonable outcomes. Yet, processed nerve allograft repair out-performed autograft repair