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A spatially distributed mathematical model is proposed for the evolution of five myocardial tissue states — healthy myocardium, inflammatory infiltrate, necrosis, replacement fibrosis, and interstitial fibrosis — in the form of a multicomponent parabolic reaction – diffusion system. At each spatial point the tissue state is described by a vector of Dirichlet concentration parameters, which allows one to encode both the expected tissue composition and the uncertainty of classification associated with transition zones of damage. For a truncated system that coincides with the original one inside an invariant region, local Lipschitz continuity and quasi-positivity of the extended reaction operator are established. Global existence of a weak solution, non-negativity, an explicit a priori upper bound, and a quantitative exponential lower bound are proved. These results imply invariance of the admissible region and hence existence of a global weak solution to the original model. Uniqueness is established in a strengthened class of solutions characterized by additional gradient regularity. For an auxiliary reversible transition matrix, a formal entropy balance is derived for smooth positive solutions; for the clinically motivated irreversible transition matrix, a modal spectral stability condition is formulated for the linearized problem. On the numerical side, a positivity condition is derived for the reaction substep of a splitting scheme relative to an adaptive upper trajectory bound, and numerical experiments are conducted for a reproducible calibrated computational variant. In the one-dimensional scenario, by day $60$ the replacement-fibrosis fraction at the lesion center increases from $0.091$ to $0.855$, while the total concentration parameter $\alpha_0$ rises from $11.0$ to $15.9$; in the two-dimensional scenario, on day $14$ the mean value of $\overline p_4^{}$ is $0.431$ in the lesion core versus $0.097$ in the healthy region. These results confirm the mathematical well-posedness of the proposed model and its applicability for the quantitative description of post-infarction scar formation and heterogeneous transition zones.

The grid-characteristic method is successfully used for solving hyperbolic systems of partial differential equations (for example, transport / acoustic / elastic equations). It allows to construct correctly algorithms on contact boundaries and boundaries of the integration domain, to a certain extent to take into account the physics of the problem (propagation of discontinuities along characteristic curves), and has the property of monotonicity, which is important for considered problems. In the cases of two-dimensional and three-dimensional problems the method makes use of a coordinate splitting technique, which enables us to solve the original equations by solving several one-dimensional ones consecutively. It is common to use up to 3-rd order one-dimensional schemes with simple splitting techniques which do not allow for the convergence order to be higher than two (with respect to time). Significant achievements in the operator splitting theory were done, the existence of higher-order schemes was proved. Its peculiarity is the need to perform a step in the opposite direction in time, which gives rise to difficulties, for example, for parabolic problems.

In this work coordinate splitting of the 3-rd and 4-th order were used for the two-dimensional hyperbolic problem of the linear elasticity. This made it possible to increase the final convergence order of the computational algorithm. The paper empirically estimates the convergence in L1 and L∞ norms using analytical solutions of the system with the sufficient degree of smoothness. To obtain objective results, we considered the cases of longitudinal and transverse plane waves propagating both along the diagonal of the computational cell and not along it. Numerical experiments demonstrated the improved accuracy and convergence order of constructed schemes. These improvements are achieved with the cost of three- or fourfold increase of the computational time (for the 3-rd and 4-th order respectively) and no additional memory requirements. The proposed improvement of the computational algorithm preserves the simplicity of its parallel implementation based on the spatial decomposition of the computational grid.