Since high order numerical methods frequently can attain accurate solutions more efficiently than low order methods, we develop and analyze new high order numerical integrators for the time discretization of ordinary and partial differential equations. Our novel methods address some of the issues surrounding high order numerical time integration, such as the difficulty of many popular methods' construction and handling the effects of disparate behaviors produce by different terms in the equations to be solved.;We are motivated by the simplicity of how Deferred Correction (DC) methods achieve high order accuracy [72, 27]. DC methods are numerical time integrators that, rather than calculating tedious coefficients for order conditions, instead construct high order accurate solutions by iteratively improving a low order preliminary numerical solution. With each iteration, an error equation is solved, the error decreases, and the order of accuracy increases. Later, DC methods were adjusted to include an integral formulation of the residual, which stabilizes the method. These Spectral Deferred Correction (SDC) methods [25] motivated Integral Deferred Corrections (IDC) methods. Typically, SDC methods are limited to increasing the order of accuracy by one with each iteration due to smoothness properties imposed by the gridspacing. However, under mild assumptions, explicit IDC methods allow for any explicit rth order Runge-Kutta (RK) method to be used within each iteration, and then an order of accuracy increase of r is attained after each iteration [18]. We extend these results to the construction of implicit IDC methods that use implicit RK methods, and we prove analogous results for order of convergence.;One means of solving equations with disparate parts is by semi-implicit integrators, handling a "fast" part implicitly and a "slow" part explicitly. We incorporate additive RK (ARK) integrators into the iterations of IDC methods in order to construct new arbitrary order semi-implicit methods, which we denote IDC-ARK methods. Under mild assumptions, we rigorously establish the order of accuracy, finding that using any rth order ARK method within each iteration gives an order of accuracy increase of r after each iteration [15]. We apply IDC-ARK methods to several numerical examples and present preliminary results for adaptive timestepping with IDC-ARK methods. Another means of solving equations with disparate parts is by operator splitting methods. We construct high order splitting methods by employing low order splitting methods within each IDC iteration. We analyze the efficiency of our split IDC methods as compared to high order split methods in [77] and also note that our construction is less tedious. Conservation of mass is proved for split IDC methods with semi-Lagrangian WENO reconstruction applied to the Vlasov-Poisson system. We include numerical results for the application of split IDC methods to constant advection, rotating, and classic plasma physics problems.;This is a preliminary, yet significant, step in the development of simple, high order numerical integrators that are designed for solving differential equations that display disparate behaviors. Our results could extend naturally to an asymptotic preserving setting or to other operator splittings.
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