A Posteriori Subcell Finite Volume Limiter for General $$P_NP_M$$ Schemes: Applications from Gasdynamics to Relativistic Magnetohydrodynamics

In this work, we consider the general family of the so called ADER $$P_NP_M$$ P N P M schemes for the numerical solution of hyperbolic partial differential equations with arbitrary high order of accuracy in space and time. The family of one-step $$P_NP_M$$ P N P M schemes was introduced in Dumbser (J Comput Phys 227:8209–8253, 2008) and represents a unified framework for classical high order Finite Volume (FV) schemes ($$N=0$$ N = 0 ), the usual Discontinuous Galerkin (DG) methods ($$N=M$$ N = M ), as well as a new class of intermediate hybrid schemes for which a reconstruction operator of degree M is applied over piecewise polynomial data of degree N with $$M>N$$ M > N . In all cases with $$M \ge N > 0 $$ M ≥ N > 0 the $$P_NP_M$$ P N P M schemes are linear in the sense of Godunov (Math. USSR Sbornik 47:271–306, 1959), thus when considering phenomena characterized by discontinuities, spurious oscillations may appear and even destroy the simulation. Therefore, in this paper we present a new simple, robust and accurate a posteriori subcell finite volume limiting strategy that is valid for the entire class of $$P_NP_M$$ P N P M schemes. The subcell FV limiter is activated only where it is needed, i.e. in the neighborhood of shocks or other discontinuities, and is able to maintain the resolution of the underlying high order $$P_NP_M$$ P N P M schemes, due to the use of a rather fine subgrid of $$2N+1$$ 2 N + 1 subcells per space dimension. The paper contains a wide set of test cases for different hyperbolic PDE systems, solved on adaptive Cartesian meshes that show the capabilities of the proposed method both on smooth and discontinuous problems, as well as the broad range of its applicability. The tests range from compressible gasdynamics over classical MHD to relativistic magnetohydrodynamics.

When mathematics meets thermal science: The simpler is the better

Mathematics is an important tool to dealing with a complex problem, but it might be too abstract and elusive to be directly applied by engineers. This paper shows mathematics should be simple but effective for thermodynamic problems, the simpler is the better. The two-scale fractal is used as an example to show the importance of application of mathematics to practical problems, and a fast-slow law is suggested to deal with many discontinuous problems by fractal calculus.

A New TVD Scheme for Gradient Smoothing Method Using Unstructured Grids

An improved [Formula: see text]-factor algorithm for implementing total variation diminishing (TVD) scheme has been proposed for the gradient smoothing method (GSM) using unstructured meshes. Different from the methods using structured meshes, for the methods using unstructured meshes, generally the upwind point cannot be clearly defined. In the present algorithm, the value of upwind point has been successfully approximated for unstructured meshes by using the GSM with different gradient smoothing schemes, including node GSM (nGSM) midpoint GSM (mGSM) and centroid GSM (cGSM). The present method has been used to solve hyperbolic partial differential equation discontinuous problems, where three classical flux limiters (Superbee, Van leer and Minmod) were used. Numerical results indicate that the proposed algorithm based on mGSM and cGSM schemes can avoid the numerical oscillation and reduce the numerical diffusion effectively. Generally the scheme based on cGSM leads to the best performance among the three proposed schemes in terms of accuracy and monotonicity.

Novel Insight into High-Order Numerical Manifold Method Using Complex Fourier Element Shape Functions in Statics and Dynamics

In this paper, the high-order numerical manifold method (HONMM) with new complex Fourier shape functions is developed for the simulation of elastostatic and elastodynamic problems. NMM uses two separate covers which give it the ability to analyze continuous and discontinuous problems in a unified way. The new shape functions are derived using constant and linear complex Fourier shape functions. These shape functions are able to satisfy exponential and trigonometric function fields in addition to polynomial ones, unlike classic Lagrange shape functions. Compared to the Lagrange shape functions, the proposed shape functions show much more accurate results with fewer degrees of freedom. The superiority of the proposed method over the conventional HONMM in static analysis is demonstrated through a special beam example. As cases of dynamic analysis, four free and forced vibration problems are illustrated. The results of the HONMM with the use of constant and linear complex Fourier shape functions are compared with the classic HONMM results and available analytical and other numerical solutions. The results show that the proposed method, even with less number of elements, is more accurate than the classic HONMM.

Numerical study on ice fragmentation by impact based on non-ordinary state-based peridynamics

Ice-structure interaction is currently one of the hot topics in engineering fields and has not been addressed. Traditional numerical methods derived from classical continuum mechanics have difficulties in solving such discontinuous problems of ice fragmentations. In the present paper, a non-ordinary state-based peridynamics formulation is presented to simulate the behavior of the ice under impact loads applied by a rigid ball. Ice is assumed as a viscoelastic-plastic material and simulated by the modified Drucker–Prager plasticity model. The failure criterion of ice is defined based on fracture toughness. A continuous contact algorithm is adopted to detect the contact between the rigid ball and ice particles. It is shown that numerical results are in good agreement with experimental data from open-literatures, and the non-ordinary state-based peridynamics model can capture the detail fragmentation features of ice under impact loads.

On two-scale dimension and its applications

Dimension or scale is everything. When a thing is observed by different scales, different results can be obtained. Two scales are enough for most of practical problems, and a new definition of a two-scale dimension instead of the fractal dimension is given to deal with discontinuous problems. Fractal theory considers a self-similarity pattern, which cannot be found in any a real problem, while the two-scale theory observes each problem with two scales, the large scale is for an approximate continuous problem, where the classic calculus can be fully applied, and on the smaller scale, the effect of the porous structure on the properties can be easily elucidated. This paper sheds a new light on applications of fractal theory to real problems.