Abstract We present a discontinuity aware quadrature (DAQ) rule and use it to develop implicit selfadaptive theta (SATh) schemes for the approximation of scalar hyperbolic conservation laws. Our SATh schemes require the solution of a system of two equations, one controlling the cell averages of the solution at the time levels, and the other controlling the spacetime averages of the solution. These quantities are used within the DAQ rule to approximate the time integral of the hyperbolic flux function accurately, even when the solution may be discontinuous somewhere over the time interval. The result is a finite volume scheme using the theta time stepping method, with theta defined implicitly (or selfadaptively). Two schemes are developed, selfadaptive theta upstream weighted (SAThup) for a monotone flux function using simple upstream stabilization, and selfadaptive theta Lax–Friedrichs (SAThLF) using the Lax–Friedrichs numerical flux. We prove that DAQ is accurate to second order when there is a discontinuity in the solution and third order when it is smooth. We prove that SAThup is unconditionally stable, provided that theta is set to be at least 1/2 (which means that SATh can be only first order accurate in general). We also prove that SAThup satisfies the maximummore »
Adaptive partitioned methods for the timeaccurate approximation of the evolutionary Stokes–Darcy system
This paper develops, analyzes and tests a timeaccurate partitioned method for the StokesDarcy equations. The method combines a time ﬁlter and Backward Euler scheme, is second order accurate and provide, at no extra complexity, an estimated the temporal error. This approach postprocesses the solutions of Backward Euler scheme by adding three lines to original codes to increase the time accuracy from ﬁrst order to second order. We prove long time stability and error estimates of Backward Euler plus time ﬁlter with constant time stepsize. Moreover, we extend the approach to variable time stepsize and construct adaptive algorithms. Numerical tests show convergence of our method and support the theoretical analysis.
 Award ID(s):
 1817542
 Publication Date:
 NSFPAR ID:
 10147672
 Journal Name:
 Computer methods in applied mechanics and engineering
 Volume:
 364
 ISSN:
 18792138
 Sponsoring Org:
 National Science Foundation
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