In general, there are several kinds of high order methods we have been developing. These include the development of highly accurate time integrators, new approaches for hyperbolic conservation laws, as well as a generalization of a new class of solvers based on the ideas for the method of lines transpose. Below we provide short descriptions along with references to the work.

Select Papers:

- Method of Lines Transpose -

Much of our recent work has focused on this area. Here we seek to generalize the ideas initially introduced by Rothe for analysis into a practical numerical method. The basic idea is to introduce an approximation in time, and then analytically invert the remainder of the operator using a Green's function. The methods we have developed are implicit, but they are as fast as explicit methods. These methods are both high order and unconditionally stable for a range of important problems. This means that these methods are accurate and we can take as large a time step as we choose without the method going unstable. We have been able to develop these ideas for a range of general problems, including the variable speed wave equation for acoustics and the case of non-linear diffusion equations. Further, we have extended these ideas to applications such as Maxwell's Equations for electromagnetics, high order phase field models and the Vlasov Equation. These methods were also designed with the GPGPUs in mind. The framework breaks the problem into pieces in such a way that it is easy to feed the large number of cores on a multi-core system. Papers on the base method are below:

- Method of Lines Transpose -

Much of our recent work has focused on this area. Here we seek to generalize the ideas initially introduced by Rothe for analysis into a practical numerical method. The basic idea is to introduce an approximation in time, and then analytically invert the remainder of the operator using a Green's function. The methods we have developed are implicit, but they are as fast as explicit methods. These methods are both high order and unconditionally stable for a range of important problems. This means that these methods are accurate and we can take as large a time step as we choose without the method going unstable. We have been able to develop these ideas for a range of general problems, including the variable speed wave equation for acoustics and the case of non-linear diffusion equations. Further, we have extended these ideas to applications such as Maxwell's Equations for electromagnetics, high order phase field models and the Vlasov Equation. These methods were also designed with the GPGPUs in mind. The framework breaks the problem into pieces in such a way that it is easy to feed the large number of cores on a multi-core system. Papers on the base method are below:

- M. Causley, A.J. Christlieb, B. Ong, L. VanGroningen, “Method of Lines Transpose: An Implicit Solution to the One Dimensional Wave Equation” , AMS – Mathematics of Computation, 83, 2763–2786, 2014
- M. Causley and A.J. Christlieb, "A-Stable Higher order schemes for the wave equation using a recursive convolution approach'' , SIAM Journal on Numerical Analysis, 52(1), 220--235, 2014
- M. Causley, H. Cho, A.J. Christlieb, D. Seal "Method of lines transpose: High order L-stable O(N) schemes for parabolic equations using successive convolution", SIAM numerical analysis, Vol. 54(3), 1635--1652, 2016
- M. Causley, A.J. Christlieb, E. Wolf, ''Method of Lines Transpose: an efficient unconditionally stable solver for wave propagation'' , Journal of Scientific Computing, Vol. 70(2), 896--921, 2017
- A.J. Christlieb, W. Guo, Y. Jiang, "Kernel Based High Order "Explicit" A-Stable Scheme for Nonlinear Degenerate Advection-Diffusion Equations" arXiv and submitted

Select Papers:

- New Methods for Fluid Models -

In this work we have been developing a class of multi derivative methods that are positivity preserving for general hyperbolic conservation laws. Further, we have been looking at ways of developing single step methods that are positivity preserving and proveably conservative. High order adaptive mesh refinement methods is an area we have explored. This work sparked our interest on time stepping methods in hyperbolic conservation laws. Further, our work with AMR reinforced the fact that AMR is easiest to implement with a single step time marching method. This fact about AMR has lead to many of the key ideas we have developed.

- New Methods for Fluid Models -

In this work we have been developing a class of multi derivative methods that are positivity preserving for general hyperbolic conservation laws. Further, we have been looking at ways of developing single step methods that are positivity preserving and proveably conservative. High order adaptive mesh refinement methods is an area we have explored. This work sparked our interest on time stepping methods in hyperbolic conservation laws. Further, our work with AMR reinforced the fact that AMR is easiest to implement with a single step time marching method. This fact about AMR has lead to many of the key ideas we have developed.

- C. Shen, J. Qiu and A.J. Christlieb, "High Order Adaptive Mesh Refinement Based on Weighted Essentially Non-Oscillatory Schemes'', Journal of Computational Physics, 230(10), 3780--3802, 2011
- A.J. Christlieb, Y. Guclu, D. Seal, "High-order multiderivative time integrators for hyperbolic conservation laws'', Journal of Scientific Computing, 60 (1), 101--140, 2014

- T. Xiong, J. Qiu, Z. Xu, A. Christlieb, "High Order Maximum Principle Preserving Semi-Lagrangian Finite Difference WENO schemes for the Vlasov Equation'', Journal of Computational Physics, 273, 618--639, 2014

- A.J. Christlieb, Y. Liu, Q. Tang, Z. Xu, "High order parametrized maximum-principle-preserving and positivity-preserving WENO schemes on unstructured meshes'', Journal of Computational Physics, 281, 334--351, 2015
- A.J. Christlieb, Y. Guclu, D. Seal, "The Picard integral formulation of weighted essentially non-oscillatory schemes'', SIAM Journal on Numerical Analysis, 53(4), 1833–1856, 2015
- A.J. Christlieb, D.C. Seal, Q. Tang, and Z. Xu, "An explicit high-order single-stage single-step positivity-preserving finite difference WENO method for the compressible Euler equations'', Journal on Scientific Computing, Dec 10th, 1--20, 2015

Select Papers:

-Novel Time Stepping Methods-

In this work we have been working to develop a new class of methods that make it easy to obtain high order while still preserving key properties. Chief among our accomplishments is a novel parallel time stepping method and the first explicit strong stability preserving method that breaks the order barrier.

-Novel Time Stepping Methods-

In this work we have been working to develop a new class of methods that make it easy to obtain high order while still preserving key properties. Chief among our accomplishments is a novel parallel time stepping method and the first explicit strong stability preserving method that breaks the order barrier.

- A.J. Christlieb, B. Ong and J. Qiu, "Integral Deferred Correction Methods Constructed with High Order Runge-Kutta Methods'', AMS--Mathematics of Computation, 79, 761--783, 2010.
- A.J. Christlieb, C.B. Macdonald and B. Ong, "Parallel High-Order Integrators'', SIAM Journal on Scientific Computing, 32, 818--835, 2010.
- A.J. Christlieb, M. Morton, B. Ong and J. Qiu, "Semi-Implicet Integral Deferred Correction Constructed with High Order Additive Runge-Kutta Methods'', Communications in Mathematical Sciences, 9(3), 879--902, 2011
- A.J. Christlieb, R Haynes, B. Ong, "A Parallel Space-Time Algorithm'',

SIAM Journal on Scientific Computing, 34(5):233-248, 2012 - A. J. Christlieb, C. B. Macdonald, B. W. Ong and R. J. Spiteri, "Revisionist Integral Deferred Correction with Adaptive Error and Stepsize Control'', Communications in Applied Mathematics and Computational Science, 10(1), 1--25, 2015
- A.J. Christlieb, S. Gottlieb, Z.J. Grant, and D.C. Seal, "Explicit Strong Stability Preserving Multistage Two-Derivative Time Stepping Schemes'', Journal of Scientific Computing, 68(3), 1-29, 2016
- M.M. Crockatt, A.J. Christlieb, C.K. Garrett, C. Hauck, "An Arbitrary-Order, Fully Implicit, Hybrid Kinetic Solver for Linear Radiative Transport Using Integral Deferred Correction", accepted Journal of Scientific Computing