Abstract We present a discontinuity aware quadrature (DAQ) rule and use it to develop implicit self-adaptive 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 space-time 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 self-adaptively). Two schemes are developed, self-adaptive theta upstream weighted (SATh-up) for a monotone flux function using simple upstream stabilization, and self-adaptive theta Lax–Friedrichs (SATh-LF) 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 SATh-up 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 SATh-up satisfies the maximum principle, and is total variation diminishing under appropriate monotonicity and boundary conditions. General flux functions require the SATh-LF scheme, so we assess its accuracy through numerical examples in one and two space dimensions. These results suggest that SATh-LF is also stable and satisfies the maximum principle (at least at reasonable Courant-Friedrichs-Lewy numbers). Compared to the solutions of finite volume schemes using Crank–Nicolson and backward Euler time stepping, SATh-LF solutions often approach the accuracy of the former, but without oscillation, and they are numerically less diffuse than the latter.
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A multipoint stress mixed finite element method for elasticity on quadrilateral grids
We develop a multipoint stress mixed finite element method for linear elasticity with weak stress symmetry on quadrilateral grids, which can be reduced to a symmetric and positive definite cell centered system. The method utilizes the lowest order Brezzi–Douglas–Marini finite element spaces for the stress and the trapezoidal quadrature rule in order to localize the interaction of degrees of freedom, which allows for local stress elimination around each vertex. We develop two variants of the method. The first uses a piecewise constant rotation and results in a cell‐centered system for displacement and rotation. The second uses a continuous piecewise bilinear rotation and trapezoidal quadrature rule for the asymmetry bilinear form. This allows for further elimination of the rotation, resulting in a cell‐centered system for the displacement only. Stability and error analysis is performed for both methods. First‐order convergence is established for all variables in their natural norms. A duality argument is employed to prove second order superconvergence of the displacement at the cell centers. Numerical results are presented in confirmation of the theory.
more » « less- Award ID(s):
- 1818775
- NSF-PAR ID:
- 10452039
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Numerical Methods for Partial Differential Equations
- Volume:
- 37
- Issue:
- 3
- ISSN:
- 0749-159X
- Format(s):
- Medium: X Size: p. 1886-1915
- Size(s):
- ["p. 1886-1915"]
- Sponsoring Org:
- National Science Foundation
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