Iterative solutions of mildly nonlinear systems

Casulli, Vincenzo; Zanolli, Paola

doi:10.1016/j.cam.2012.02.042

Cited by 54 publications

(78 citation statements)

References 7 publications

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“…Given the unprecedented resolution capabilities shown in various test cases, we are convinced that the method presented in this paper belongs to a new generation of shock capturing schemes for computational fluid dynamics. Future work may concern the extension of the present space-time adaptive algorithm to high order semi-implicit DG schemes, following the ideas outlined in [42,112,111,[20][21][22]24,25,45].…”

Section: Resultsmentioning

confidence: 99%

Space–time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting

et al. 2015

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A posteriori sub-cell finite volume limiter MOOD paradigm High order space-time adaptive mesh refinement (AMR) ADER-DG and ADER-WENO finite volume schemes Hyperbolic conservation laws a b s t r a c t In this paper we present a novel arbitrary high order accurate discontinuous Galerkin (DG) finite element method on space-time adaptive Cartesian meshes (AMR) for hyperbolic conservation laws in multiple space dimensions, using a high order a posteriori sub-cell ADER-WENO finite volume limiter. Notoriously, the original DG method produces strong oscillations in the presence of discontinuous solutions and several types of limiters have been introduced over the years to cope with this problem. Following the innovative idea recently proposed in [53], the discrete solution within the troubled cells is recomputed by scattering the DG polynomial at the previous time step onto a suitable number of sub-cells along each direction. Relying on the robustness of classical finite volume WENO schemes, the sub-cell averages are recomputed and then gathered back into the DG polynomials over the main grid. In this paper this approach is implemented for the first time within a space-time adaptive AMR framework in two and three space dimensions, after assuring the proper averaging and projection between sub-cells that belong to different levels of refinement. The combination of the sub-cell resolution with the advantages of AMR allows for an unprecedented ability in resolving even the finest details in the dynamics of the fluid. The spectacular resolution properties of the new scheme have been shown through a wide number of test cases performed in two and in three space dimensions, both for the Euler equations of compressible gas dynamics and for the magnetohydrodynamics (MHD) equations.

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Section: Resultsmentioning

confidence: 99%

Space–time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting

et al. 2015

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“…The system for the pressure (49) is again a mildly nonlinear system of the form (23) with a linear part that is symmetric and at least positive semidefinite. 39,40 Note that in the incompressible limit M → 0, following the asymptotic analysis performed in other works, 100-104 the pressure tends to a constant and the contribution of the kinetic energy k can be neglected w.r.t. 39,40 Note that in the incompressible limit M → 0, following the asymptotic analysis performed in other works, 100-104 the pressure tends to a constant and the contribution of the kinetic energy k can be neglected w.r.t.…”

Section: Pressure Subsystemmentioning

confidence: 89%

“…Therefore, the application of semi-implicit methods to compressible flows with shock waves is still quite rare, and some recent developments in this direction have been made only very recently in other works, [26][27][28][29][30][31] where new conservative pressure-based semi-implicit schemes have been proposed that are also suitable for the simulation of flow problems including shock waves. The properties of the pressure system allow the use of the Newton-type techniques of Casulli et al, [37][38][39][40] for which convergence has been rigorously proven. The existing semi-implicit schemes for MHD either apply only to the incompressible or anelastic case, or they are not based on a conservative formulation, see, eg, other works.…”

Section: Introductionmentioning

confidence: 99%

A divergence‐free semi‐implicit finite volume scheme for ideal, viscous, and resistive magnetohydrodynamics

Dumbser

Balsara

Tavelli

et al. 2018

Numerical Methods in Fluids

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Summary In this paper, we present a novel pressure‐based semi‐implicit finite volume solver for the equations of compressible ideal, viscous, and resistive magnetohydrodynamics (MHD). The new method is conservative for mass, momentum, and total energy, and in multiple space dimensions, it is constructed in such a way as to respect the divergence‐free condition of the magnetic field exactly, also in the presence of resistive effects. This is possible via the use of multidimensional Riemann solvers on an appropriately staggered grid for the time evolution of the magnetic field and a double curl formulation of the resistive terms. The new semi‐implicit method for the MHD equations proposed here discretizes the nonlinear convective terms as well as the time evolution of the magnetic field explicitly, whereas all terms related to the pressure in the momentum equation and the total energy equation are discretized implicitly, making again use of a properly staggered grid for pressure and velocity. Inserting the discrete momentum equation into the discrete energy equation then yields a mildly nonlinear symmetric and positive definite algebraic system for the pressure as the only unknown, which can be efficiently solved with the (nested) Newton method of Casulli et al. The pressure system becomes linear when the specific internal energy is a linear function of the pressure. The time step of the scheme is restricted by a CFL condition based only on the fluid velocity and the Alfvén wave speed and is not based on the speed of the magnetosonic waves. Being a semi‐implicit pressure‐based scheme, our new method is therefore particularly well suited for low Mach number flows and for the incompressible limit of the MHD equations, for which it is well known that explicit density‐based Godunov‐type finite volume solvers become increasingly inefficient and inaccurate because of the more and more stringent CFL condition and the wrong scaling of the numerical viscosity in the incompressible limit. We show a relevant MHD test problem in the low Mach number regime where the new semi‐implicit algorithm is a factor of 50 faster than a traditional explicit finite volume method, which is a very significant gain in terms of computational efficiency. However, our numerical results confirm that our new method performs well also for classical MHD test cases with strong shocks. In this sense, our new scheme is a true all Mach number flow solver.

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“…, n and that A satisfies either one of the following properties: A1: A is a Stieltjes matrix, i.e., a symmetric M-matrix (for more details on M-matrix, we refer to [12,19,20] PLS (1.1) arises from the discretization of the free-surface hydrodynamic problem and a suitable, implicit discretization of this problem yields a piecewise linear system of the form (1.1) with A satisfying A1 or A2 at each time step (see, e.g., [2,3,5,6], for more details). Therefore, it is meaningful to develop efficient algorithms to solve PLS (1.1) and it has attracted much attention (see, e.g., [2][3][4]6,7,9,17,18]). …”

mentioning

confidence: 99%

A damped semismooth Newton method for the Brugnano–Casulli piecewise linear system

Sun

Liu

2014

Bit Numer Math

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Iterative solutions of mildly nonlinear systems

Cited by 54 publications

References 7 publications

Space–time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting

Space–time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting

A divergence‐free semi‐implicit finite volume scheme for ideal, viscous, and resistive magnetohydrodynamics

A damped semismooth Newton method for the Brugnano–Casulli piecewise linear system

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