Third-order Accurate Method Research Articles

A Third-Order Accurate Wave Propagation Algorithm for Hyperbolic Partial Differential Equations

We extend LeVeque’s wave propagation algorithm, a widely used finite volume method for hyperbolic partial differential equations, to a third-order accurate method. The resulting scheme shares main properties with the original method, i.e., it is based on a wave decomposition at grid cell interfaces, it can be used to approximate hyperbolic problems in divergence form as well as in quasilinear form and limiting is introduced in the form of a wave limiter.

Development of a Numerical Method for Analyzing Fire Plume Equations Using an Improved Version of a Quasi-Third-Order Accurate CIP Method That Eliminates Numerical Oscillation

This study is concerned with simulations of fire phenomena using field equation models. When performing numerical computation of a fire plume accompanied by gas and smoke, the phenomena are expressed through simultaneous non-stationary nonlinear second-order partial differential equations containing advective terms. Those advective terms are then subjected to difference approximations. The ordinary CIP method, which allows us to find an approximate solution using a third-order interpolation function, is often influenced by numerical oscillations emerging specifically with the third-order accuracy, thereby drastically reducing the numerical stability. We developed a new computational algorithm called mCIP method to eliminate these numerical oscillations generated in the numerical computation of partial differential equations when a third-order accurate method is used. A characteristic feature of this mCIP is that it is based on the first-order upwind difference scheme, which does not generate numerical oscillations in those regions where such oscillations may normally occur. We successfully evaluated the performance of this mCIP method and verified its effectiveness by comparing the results of computations with other representative methods, such as CIP method, TVD method and first-order accurate difference method.

High-Resolution Conservative Algorithms for Advection in Incompressible Flow

A class of high-resolution algorithms is developed for advection of a scalar quantity in a given incompressible flow field in one, two, or three space dimensions. Multidimensional transport is modeled using a wave-propagation approach in which the flux at each cell interface is built up on the basis of information propagating in the direction of this interface from neighboring cells. A high-resolution second-order method using slope limiters is quite easy to implement. For constant flow, a minor modification gives a third-order accurate method. These methods are stable for Courant numbers up to 1. Fortran implementations are available by anonymous ftp.