Iterative Approximate Factorization of Difference Operators of High-Order Accurate Bicompact Schemes for Multidimensional Nonhomogeneous Quasilinear Hyperbolic Systems

Bicompact schemes for multidimensional hyperbolic equations on Cartesian meshes with solution-based AMR

“…In scheme (2), for example, Newton's method or iterated approximate factorization [26] is used for this purpose. Newton's method is applied in scheme (3).…”

Two variants of parallel implementation of high-order accurate bicompact schemes for multi-dimensional inhomogeneous transport equation

“…Note that the convergence of the iterative approximate factorization method for bicompact schemes as applied to the nonstationary 2D linear homogeneous transport equation was proved in [9,10].…”

“…For example, Intel processors of the last Xeon Phi generation have 61 to 72 cores per processor, and up to four threads can be executed on each core. Moreover, such processors possess low efficiency and storage as compared with processors having fewer cores (8)(9)(10)(11)(12). Therefore, to use the capabilities of such processors to a full extent, we need algorithms that are well scalable on small-sized problems.…”

“…Later, bicompact schemes were constructed for the two-dimensional (2D) and three-dimensional (3D) nonstationary inhomogeneous linear transport equations [6,7] and for systems of nonstationary multidimensional quasilinear hyperbolic equations [8]. Note that, in bicompact schemes [8,9] for quasilinear hyperbolic equations, an increase in the order of accuracy is ensured not with the help of auxiliary integral averages of the sought function over grid cells, but rather with the help of auxiliary values of the sought function at half-integer spatial grid nodes. To find these quantities, additional difference equations are derived that approximate consequences of the basic differential equations.…”

“…It is based on an approximate factorization of multidimensional difference operators. This algorithm makes use of iterations in order to preserve the high (higher than the second) order of accuracy of the schemes in time [9,10]. We compare the efficiency of two parallel algorithms for computing the equations of multidimensional bicompact schemes, namely, the space marching algorithm for unfactorized schemes and an algorithm for computing factorized schemes.…”