MSeeker1340

using OrdinaryDiffEq, ExponentialUtilities
using LinearAlgebra, SparseArrays

# Example: Use expv to solve constant-coefficient linear systems
#   u' = ϵu_xx + u, u ∈ [0, 1], t ∈ [0, 1], Dirichlet BC
# The analytical solution is
#   u(t) = exp(tA)u(0), A = L + I
# which we approximate with expv/expv_timestep
N = 100; T = 1.0
dx = 1 / (N + 1)

MSeeker1340

During the summer, I contributed to multiple repositories under the JuliaDiffEq organization and completed the implementation of Exponential Runge-Kutta (ExpRK) integrators. My work can be roughly summarized into three parts: the exponential utilities, the integrators and sparse Jacobian support.

Exponential Utilities

The development of utility functions/types used by the exponential occurs on the first four weeks and is summarized by the following blog posts:

• Week 1 and 2
• Week 3 and 4

Near the end of GSoC, I also migrated the utilities to a separate package ExponentialUtilities under JuliaDiffEq.

MSeeker1340

include("Phi.jl")
using Phi: phim_alt, phimc

Arnoldi iteration

The Arnoldi iteratrion algorithm computes a basis for the Krylov subspace

MSeeker1340

include("Phi.jl")
using Phi
using PyPlot

Utilizing the Sidje formula

MSeeker1340

Phi functions

The phi functions can be defined using the recurrence relation

$$ \varphi_{k+1}(z) = \frac{\varphi_k(z) - \varphi_k(0)}{z},\quad \varphi_0(z) = e^z $$

The zero point values are $\varphi_k(0) = 1/k!$.

The recurrence relation naturally lends itself to a dynamic programming style implementation, where it is more efficient to calculate all $\varphi_i(z)$ for $0 \le i \le k$.