Features

• Provides a simple syntactic language to specify a PDE and its boundary conditions over a discretized numerical domain (Using a derived Maple data structure: Discrete Domain Specifier DDS)
• Creates Fortran routines and C wrappers for a given DDS that can evaluate an FDA expression or solve it using Newton-Gauss-Seidel iterative method (commonly used for non-linear PDEs)
• Allows user specified discretization scheme such as forward, backward, centered or asymmetric discretization with used-defined accuracy via a data type (Maple table): Finite Difference Specifier, FDS.
• Can handle various boundary conditions such as periodic boundary, inner boundary conditions (imposing specific symmetry on the functions such as even or odd behaviour on axis/point of symmetry) or other types of user specified conditions, such as particular asymptotic behaviour, outgoing (Neumann), fixed (Dirichlet), etc.
• Generates finite difference approximation of an arbitrary partial differential expression of up to 4 variables, f(t,x,y,z), for a used defined accuracy and discretization scheme.
• Creates parallel ready Fortran routine that can interface with parallelization infrastructures such as PAMR
• Can parse a given continuum algebraic/differential expression and can convert it directly to a Fortran routine to evaluate it on the discretized domain. Doing so, it provides a rapid prototyping language to create residual evaluators for a given differential expression for testing purposes.
• Allows manual overwrite of discretization scheme by creating Manual Finite Difference Operators defined via the built-in fundamental difference operator.
• Mainly designed with a focus on highly complex and non-linear time dependent PDEs or boundary value PDEs that occur in physical systems.
• It is written in Maple, a powerful symbolic manipulation language and therefore inherits all the capabilities of Maple, including various tools to deal with PDEs and algebraic expressions.
• It was originally developed in Numerical Relativity group in UBC to solve the Einstein's equation which is a set of 10 highly non-linear coupled PDEs, where even writing down the equations in their continuum form needs to be done using symbolic manipulation software such as GRTensor . Therefore it is mostly developed to deal with large differential expressions that are machine generated and carries a full explicit form with all the dependencies and derivatives written in Maple's canonical form like: diff(f(t,x,y),x,x)+g(x,y)+diff(g(x,y),y,y,y)+...
• Being a Maple toolbox, FD unifies the two parts of: (1)deriving and manipulating the set of PDEs with all the required work variables, and (2) converting them into FDA expression and solving them. This can further help to reduce potential human errors.