| 1. |
|
|
| 2. |
- Friedrich, Lucas, et al.
(författare)
-
An Entropy Stable h/p Non-Conforming Discontinuous Galerkin Method with the Summation-by-Parts Property
- 2018
-
Ingår i: Journal of Scientific Computing. - : Springer. - 0885-7474 .- 1573-7691. ; 77:2, s. 689-725
-
Tidskriftsartikel (refereegranskat)abstract
- This work presents an entropy stable discontinuous Galerkin (DG) spectral element approximation for systems of non-linear conservation laws with general geometric (h) and polynomial order (p) non-conforming rectangular meshes. The crux of the proofs presented is that the nodal DG method is constructed with the collocated Legendre–Gauss–Lobatto nodes. This choice ensures that the derivative/mass matrix pair is a summation-by-parts (SBP) operator such that entropy stability proofs from the continuous analysis are discretely mimicked. Special attention is given to the coupling between non-conforming elements as we demonstrate that the standard mortar approach for DG methods does not guarantee entropy stability for non-linear problems, which can lead to instabilities. As such, we describe a precise procedure and modify the mortar method to guarantee entropy stability for general non-linear hyperbolic systems on h / p non-conforming meshes. We verify the high-order accuracy and the entropy conservation/stability of fully non-conforming approximation with numerical examples.
|
|
| 3. |
- Friedrich, Lucas, et al.
(författare)
-
Conservative and stable degree preserving SBP operators for non-conforming meshes
- 2018
-
Ingår i: Journal of Scientific Computing. - : Springer-Verlag New York. - 0885-7474 .- 1573-7691. ; 75:2, s. 657-686
-
Tidskriftsartikel (refereegranskat)abstract
- Non-conforming numerical approximations offer increased flexibility for applications that require high resolution in a localized area of the computational domain or near complex geometries. Two key properties for non-conforming methods to be applicable to real world applications are conservation and energy stability. The summation-by-parts (SBP) property, which certain finite-difference and discontinuous Galerkin methods have, finds success for the numerical approximation of hyperbolic conservation laws, because the proofs of energy stability and conservation can discretely mimic the continuous analysis of partial differential equations. In addition, SBP methods can be developed with high-order accuracy, which is useful for simulations that contain multiple spatial and temporal scales. However, existing non-conforming SBP schemes result in a reduction of the overall degree of the scheme, which leads to a reduction in the order of the solution error. This loss of degree is due to the particular interface coupling through a simultaneous-approximation-term (SAT). We present in this work a novel class of SBP-SAT operators that maintain conservation, energy stability, and have no loss of the degree of the scheme for non-conforming approximations. The new degree preserving discretizations require an ansatz that the norm matrix of the SBP operator is of a degree ≥ 2p, in contrast to, for example, existing finite difference SBP operators, where the norm matrix is 2p − 1 accurate. We demonstrate the fundamental properties of the new scheme with rigorous mathematical analysis as well as numerical verification.
|
|
| 4. |
- Friedrich, Lucas, et al.
(författare)
-
Entropy Stable Space-Time Discontinuous Galerkin Schemes with Summation-by-Parts Property for Hyperbolic Conservation Laws
- 2019
-
Ingår i: Journal of Scientific Computing. - : Springer. - 0885-7474 .- 1573-7691. ; 80:1, s. 175-222
-
Tidskriftsartikel (refereegranskat)abstract
- This work examines the development of an entropy conservative (for smooth solutions) or entropy stable (for discontinuous solutions) space-time discontinuous Galerkin (DG) method for systems of nonlinear hyperbolic conservation laws. The resulting numerical scheme is fully discrete and provides a bound on the mathematical entropy at any time according to its initial condition and boundary conditions. The crux of the method is that discrete derivative approximations in space and time are summation-by-parts (SBP) operators. This allows the discrete method to mimic results from the continuous entropy analysis and ensures that the complete numerical scheme obeys the second law of thermodynamics. Importantly, the novel method described herein does not assume any exactness of quadrature in the variational forms that naturally arise in the context of DG methods. Typically, the development of entropy stable schemes is done on the semidiscrete level ignoring the temporal dependence. In this work, we demonstrate that creating an entropy stable DG method in time is similar to the spatial discrete entropy analysis, but there are important (and subtle) differences. Therefore, we highlight the temporal entropy analysis throughout this work. For the compressible Euler equations, the preservation of kinetic energy is of interest besides entropy stability. The construction of kinetic energy preserving (KEP) schemes is, again, typically done on the semidiscrete level similar to the construction of entropy stable schemes. We present a generalization of the KEP condition from Jameson to the space-time framework and provide the temporal components for both entropy stability and kinetic energy preservation. The properties of the space-time DG method derived herein are validated through numerical tests for the compressible Euler equations. Additionally, we provide, in appendices, how to construct the temporal entropy stable components for the shallow water or ideal magnetohydrodynamic (MHD) equations.
|
|
| 5. |
|
|
| 6. |
- Neyroud, D, et al.
(författare)
-
Toxic doses of caffeine are needed to increase skeletal muscle contractility
- 2019
-
Ingår i: American journal of physiology. Cell physiology. - : American Physiological Society. - 1522-1563 .- 0363-6143. ; 316:2, s. C246-C251
-
Tidskriftsartikel (refereegranskat)abstract
- Discrepant results have been reported regarding an intramuscular mechanism underlying the ergogenic effect of caffeine on neuromuscular function in humans. Here, we reevaluated the effect of caffeine on muscular force production in humans and combined this with measurements of the caffeine dose-response relationship on force and cytosolic free [Ca2+] ([Ca2+]i) in isolated mouse muscle fibers. Twenty-one healthy and physically active men (29 ± 9 yr, 178 ± 6 cm, 73 ± 10 kg, mean ± SD) took part in the present study. Nine participants were involved in two experimental sessions during which supramaximal single and paired electrical stimulations (at 10 and 100 Hz) were applied to the femoral nerve to record evoked forces. Evoked forces were recorded before and 1 h after ingestion of 1) 6 mg caffeine/kg body mass or 2) placebo. Caffeine plasma concentration was measured in 12 participants. In addition, submaximal tetanic force and [Ca2+]iwere measured in single mouse flexor digitorum brevis (FDB) muscle fibers exposed to 100 nM up to 5 mM caffeine. Six milligrams of caffeine per kilogram body mass (plasma concentration ~40 µM) did not increase electrically evoked forces in humans. In superfused FDB single fibers, millimolar caffeine concentrations (i.e., 15- to 35-fold above usual concentrations observed in humans) were required to increase tetanic force and [Ca2+]i. Our results suggest that toxic doses of caffeine are required to increase muscle contractility, questioning the purported intramuscular ergogenic effect of caffeine in humans.
|
|
| 7. |
- Winters, Andrew Ross, et al.
(författare)
-
A comparative study on polynomial dealiasing and split form discontinuous Galerkin schemes for under-resolved turbulence computations
- 2018
-
Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 372, s. 1-21
-
Tidskriftsartikel (refereegranskat)abstract
- This work focuses on the accuracy and stability of high-order nodal discontinuous Galerkin (DG) methods for under-resolved turbulence computations. In particular we consider the inviscid Taylor-Green vortex (TGV) flow to analyse the implicit large eddy simulation (iLES) capabilities of DG methods at very high Reynolds numbers. The governing equations are discretised in two ways in order to suppress aliasing errors introduced into the discrete variational forms due to the under-integration of non-linear terms. The first, more straightforward way relies on consistent/over-integration, where quadrature accuracy is improved by using a larger number of integration points, consistent with the degree of the non-linearities. The second strategy, originally applied in the high-order finite difference community, relies on a split (or skew-symmetric) form of the governing equations. Different split forms are available depending on how the variables in the non-linear terms are grouped. The desired split form is then built by averaging conservative and non-conservative forms of the governing equations, although conservativity of the DG scheme is fully preserved. A preliminary analysis based on Burgers’ turbulence in one spatial dimension is conducted and shows the potential of split forms in keeping the energy of higher-order polynomial modes close to the expected levels. This indicates that the favourable dealiasing properties observed from split-form approaches in more classical schemes seem to hold for DG. The remainder of the study considers a comprehensive set of (under-resolved) computations of the inviscid TGV flow and compares the accuracy and robustness of consistent/over-integration and split form discretisations based on the local Lax-Friedrichs and Roe-type Riemann solvers. Recent works showed that relevant split forms can stabilize higher-order inviscid TGV test cases otherwise unstable even with consistent integration. Here we show that stable high-order cases achievable with both strategies have comparable accuracy, further supporting the good dealiasing properties of split form DG. The higher-order cases achieved only with split form schemes also displayed all the main features expected from consistent/over-integration. Among test cases with the same number of degrees of freedom, best solution quality is obtained with Roe-type fluxes at moderately high orders (around sixth order). Solutions obtained with very high polynomial orders displayed spurious features attributed to a sharper dissipation in wavenumber space. Accuracy differences between the two dealiasing strategies considered were, however, observed for the low-order cases, which also yielded reduced solution quality compared to high-order results.
|
|
