Combining DPG in space with DPG time-marching scheme for the transient advection-reaction equation

Technical Report · 01 May 2022

DOI:https://doi.org/10.2172/1869510· OSTI ID:1869510

Munoz-Matute, Judit ^[1]; Demkowicz, Leszek ^[2]; Roberts, Nathaniel David ^[3]

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1. Basque Center for Applied Mathematics (BCAM), Bilbao (Spain); Univ. of Texas, Austin, TX (United States)
2. Univ. of Texas, Austin, TX (United States)
3. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

In this article, we present a general methodology to combine the Discontinuous PetrovGalerkin (DPG) method in space and time in the context of methods of lines for transient advection-reaction problems. We first introduce a semidiscretization in space with a DPG method redefining the ideas of optimal testing and practicality of the method in this context. Then, we apply the recently developed DPG-based time-marching scheme, which is of exponential-type, to the resulting system of Ordinary Differential Equations (ODEs). We also discuss how to efficiently compute the action of the exponential of the matrix coming from the space semidiscretization without assembling the full matrix. Finally, we verify the proposed method for 1D+time advection-reaction problems showing optimal convergence rates for smooth solutions and more stable results for linear conservation laws comparing to the classical exponential integrators.

• Research Organization: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)
• Sponsoring Organization: USDOE National Nuclear Security Administration (NNSA)
• DOE Contract Number: NA0003525
• OSTI ID: 1869510
• Report Number(s): SAND2022-7080R; 706732
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
DPG method
advection-reaction equation
exponential integrators
method of lines
optimal test functions
ultraweak formulation