36,192 research outputs found
Time Integration Methods of Fundamental Solutions and Approximate Fundamental Solutions for Nonlinear Elliptic Partial Differential Equations
A time-dependent method is coupled with the Method of Approximate Particular Solutions (MAPS) of Delta-shaped basis functions, the Method of Fundamental Solutions (MFS), and the Method of Approximate Fundamental Solutions (MAFS) to solve a second order nonlinear elliptic partial differential equation (PDE) on regular and irregular shaped domains. The nonlinear PDE boundary value problem is first transformed into a time-dependent quasilinear problem by introducing a fictitious time. Forward Euler integration is then used to ultimately convert the problem into a sequence of time-dependent linear nonhomogeneous modified Helmholtz boundary value problems on which the superposition principle is applied to split the numerical solution at each time step into a homogeneous solution and an approximate particular solution. The Crank-Nicholson method is also examined as an option for the numerical integration as opposed to the forward Euler method. A Delta-shaped basis function, which can handle scattered data in various domains, is used to provide an approximation of the source function at each time step and allows for a derivation of an approximate particular solution of the associated nonhomogeneous equation using the MAPS. The corresponding homogeneous boundary value problem is solved using MFS or MAFS. Numerical results support the accuracy and validity of these computational methods. The proposed numerical methods are additionally applied in nonlinear thermal explosion to determine the steady state critical condition in explosive regimes
Supersonic Flow onto Solid Wedges, Multidimensional Shock Waves and Free Boundary Problems
When an upstream steady uniform supersonic flow impinges onto a symmetric
straight-sided wedge, governed by the Euler equations, there are two possible
steady oblique shock configurations if the wedge angle is less than the
detachment angle -- the steady weak shock with supersonic or subsonic
downstream flow (determined by the wedge angle that is less or larger than the
sonic angle) and the steady strong shock with subsonic downstream flow, both of
which satisfy the entropy condition. The fundamental issue -- whether one or
both of the steady weak and strong shocks are physically admissible solutions
-- has been vigorously debated over the past eight decades. In this paper, we
survey some recent developments on the stability analysis of the steady shock
solutions in both the steady and dynamic regimes. For the static stability, we
first show how the stability problem can be formulated as an initial-boundary
value type problem and then reformulate it into a free boundary problem when
the perturbation of both the upstream steady supersonic flow and the wedge
boundary are suitably regular and small, and we finally present some recent
results on the static stability of the steady supersonic and transonic shocks.
For the dynamic stability for potential flow, we first show how the stability
problem can be formulated as an initial-boundary value problem and then use the
self-similarity of the problem to reduce it into a boundary value problem and
further reformulate it into a free boundary problem, and we finally survey some
recent developments in solving this free boundary problem for the existence of
the Prandtl-Meyer configurations that tend to the steady weak supersonic or
transonic oblique shock solutions as time goes to infinity. Some further
developments and mathematical challenges in this direction are also discussed.Comment: 19 pages; 8 figures; accepted by Science China Mathematics on
February 22, 2017 (invited survey paper). doi: 10.1007/s11425-016-9045-
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