1. Partial Differential Equations (PDEs) classification groups
PDEs can be classified from different perspectives:
Order of PDE: The highest order of PDE
Number of variables: The number of independent variables for all the involved functions: Note: Under certain conditions when initial / boundary conditions and partial derivatives are independent of a given variable we can reduce the number of variables.
Source: [Farlow, 2012]

2. Partial Differential Equations (PDEs) classification groups
3. Homogeneity: If the source term (right hand side) of the equation is zero the PDE is called homogeneous. The same concept applies to initial (IC) and boundary (BC) conditions of a PDE (RHS of the IC/BC differential operator is zero)
Type of coefficient:
Constant coefficient (function & its derivative terms have constant coefficients)
Variable coefficient
Coefficients only function of independent variables (e.g. x, t)
Coefficients function of independent variables AND the function (e.g. x, t, u)
Source: [Farlow, 2012]

3. Partial Differential Equations (PDEs) classification groups
Hyperbolic / parabolic / elliptic PDEs: - The classification becomes more clear in the next few slides. Below is the brief description of some of their characteristics and sample applications.
Hyperbolic PDEs correspond to the propagation of waves and there is a finite speed of propagation of waves. They tend to preserve or generate discontinuities (in the absence of damping). Hyperbolic PDEs are often transient although some steady-state limits of transient PDEs can be hyperbolic as well (e.g. steady advection problems). Examples: Elastodynamics, Transient electromagnetics; Acoustic equation.
Parabolic PDEs: Unlike hyperbolic PDEs the speed of propagation of information is infinite for parabolic PDEs. They also tend to dissipate sharp solution features and have a “diffusive” behavior. Many transient diffusion problems are modeled (or idealized) by parabolic PDEs. Some examples are Examples: Fourier heat equation; Viscous flow (Navier-Stokes equations)
Elliptic PDEs: Elliptic problems are characterized by the global coupling of the solution. They often correspond to steady-state limit of hyperbolic and parabolic PDEs.
Source: [Farlow, 2012]

4. Types of PDEs Elliptic, parabolic, hyperbolic

5. Types of PDEs Elliptic, parabolic, hyperbolic
Burger’s equation (nonlinear)
t = 0, smooth solution
t > 0, shock has formed

6. Partial Differential Equations (PDEs) classification groups
6. Linearity: - The PDE is linear if the dependent variable and all its derivatives appear linearly in the PDE. The nonlinear PDEs are classified into several groups as their solution characteristics can be quite distinct:
Notations:
Multi-index
Multi-index partial derivative
example:
Collection of all partial derivatives of order k:
Source: [Levandosky, 2002]

7. Partial Differential Equations (PDEs) llinear/nonlinear classifications
General form of PDE:
Linear nonlinear (in order)
Linear: If u and its derivatives appear in a linear fashion. That is F can be written as, Examples:
Source: [Levandosky, 2002]

8. Partial Differential Equations (PDEs) llinear/nonlinear classifications
Semi-linear: is a nonlinear PDE where the highest order derivatives can be written in a linear fashion of functions of x. That is, such coefficients are only functions of independent coordinate x. The PDE can be written as, Examples:
Source: [Levandosky, 2002]

9. Partial Differential Equations (PDEs) llinear/nonlinear classifications
Quasi-linear: is a nonlinear PDE, that is not semilinear and its highest derivatives can be written as linear function of functions of x and lower order derivatives of u. That is, it can be written as, That is the coefficients of highest order terms depend on
Examples:
Is quasilinear
semilinear if a(x, y), b(x,y)
linear if a(x, y), b(x, y) and c = u d(x, y)
Source: [Levandosky, 2002]

10. Partial Differential Equations (PDEs) llinear/nonlinear classifications
D. Fully-nonlinear: If it’s nonlinear and cannot be written in quasi-linear, semi-linear forms.
For a list of well-known nonlinear (semi-linear, quasi-linear, and fully nonlinear) PDEs refer to here (
Source: [Levandosky, 2002]

11. Partial Differential Equations (PDEs) classification groups
Semi-linear
Quasi-linear
Fully-nonlinear
Source: [Farlow, 2012]

12. Solution to 1D wave equation
We are interested in solving the 1st order linear PDE in two variables:
The idea is to express the two partial derivatives as one derivative term:
Thus the PDE turns to an ODE:
Source: [Farlow, 2012]

13. Solution to 1D wave equation
The ODEs are solved along characteristic where variable s changes
Solution is obtained by taking points from IC (and BC) and solving for the solution along the characteristic
Source: [Farlow, 2012]

14. Solution to 1D wave equation
How this is done? By using the chain rule:
The solution to ODEs provides the direction of characteristics
where x and t are obtained from ODEs:
Source: [Farlow, 2012]

15. Solution to 1D wave equation Example: constant coefficients
Consider the following PDE with constant coefficients:
The characteristic equations become:
c1 and c2 can be obtained from initial condition
- At the start of characteristic lines x is equal to a secondary variable t (it can also be set to x)
- Time is equal to zero at the beginning of characteristic lines
Source: [Farlow, 2012]

16. Solution to 1D wave equation Example: constant coefficients
Time is equal to zero at the beginning of characteristic lines
Note: We can simply write x(0) = x and not introduce the intermediate parameter t.
Source: [Farlow, 2012]

17. Solution to 1D wave equation Example: constant coefficients
Continuation, the solution of PDE
And characteristic equations
with follow as:
We note that and characteristics look like
Source: [Farlow, 2012]

18. Solution to 1D wave equation Example: constant coefficients
Thus the PDE turns to the following ODE:
Solving the Initial Value Problem (IVP) we get
and by inverting (s, t)  (x, t) we get
Thus becomes
Source: [Farlow, 2012]

19. Solution to 1D wave equation Example: constant coefficients
So the solution to Is
Which is a sine wave moving with speed 1 and damping in time
Source: [Farlow, 2012]

20. Solution to 1D wave equation Example: variable coefficients
Consider the problem
Step 1: The ODEs for characteristics are:
Hence
Source: [Farlow, 2012]

21. Solution to 1D wave equation Example: variable coefficients
Step 2: we solve the ODE
we IC
Which gives
Step 3: By inverting
we get
and by plugging back into we get the final solution
e.g. for initial condition F(x) = sin(x) we have
Source: [Farlow, 2012]

22. Solution to 1D wave equation Discussion on hyperbolicity
Key aspect of the first order PDEs we discussed was the solution for
characteristic curves along which the solution could be obtained by the solution of an ODE.
The spatial to temporal slope of characteristics corresponds to the wave speed.
The hyperbolicity of a PDE corresponds to having characteristic curves along which the solution propagates. For higher order PDEs we investigate if we can breakdown the PDEs into the solution of ODEs along characteristic curves. If this is possible the PDE is hyperbolic and has a finite speed of information propagation at a given point. 1
C, wave speed

23. Classification of second order PDEs: Two independent parameters
Consider a general 2nd order PDE
We restrict our attention to linear PDE with 2 independent parameters below (results can easily be generalized to semi-linear case):
where A, B, C, D, E, F, G are functions of (x, y) in general (linear PDE). The classification of PDE at a given point (x0, y0) is as follows:
If 1 holds for all (x, y) the PDE is called hyperbolic for all positions (same for 2 and 3)

24. Classification of second order PDEs: Two independent parameters
Examples:
A = 1, B = 0, C = -1 
B2 – 4AC = 4 > 0
A = 1, B = 0, C = 1 
B2 – 4AC = -4 < 0
A = 0, B = 0, C = -1 
B2 – 4AC = 0

25. Classification of second order PDEs: Canonical form
The idea is to cast the PDE in the canonical form
Source: [Farlow, 2012]

26. Classification of second order PDEs: Canonical form
We look for parameters x and h that cast the PDE into the hyperbolic form
By transformation
By change of parameters we obtain
Source: [Farlow, 2012]

27. Classification of second order PDEs: Canonical form
Substituting into the original PDE
we obtain
where
To cast it in the form we need to set A and C to zero.
Source: [Farlow, 2012]

28. Classification of second order PDEs: Canonical form
Which results in equations of the form
The solutions to these equations are:
He have three cases:
B2 – 4AC > 0 : Two distinct values for xx/ xy and hx/ hy. We can cast the equation in hyperbolic canonical form.
B2 – 4AC = 0 : ONLY one distinct values for xx/ xy and hx/ hy. We cannot cast the equation in hyperbolic form, but can cast in parabolic form.
B2 – 4AC < 0 : NO REAL roots xx/ xy and hx/ hy. We cannot cast the equation in hyperbolic canonical form, but can cast in elliptic form.

29. Classification of second order PDEs: Characteristic values and curves
x and h are called the characteristic parameters (similar to the first ordr PDE)
By solving the previous page 2nd order equation we can find how the contour lines (constant values) for x and h look like in (x, y) space.

30. Classification of second order PDEs: Example
A constant coefficient hyperbolic example:
The equations for characteristic curves are:
After integration we obtain
By leaving x and y on the RHS of equation we obtain equations for x and h:

31. Classification of second order PDEs: Example
And the characteristic curves look like

32. Classification of second order PDEs: Example with variable coefficients
Consider the PDE
which is a hyperbolic equation in the first quadrant.
We find the characteristics by the equation
By solving these equations and moving x, y to the RHS we obtain

33. Classification of second order PDEs: Example with variable coefficients
To obtain the form of the equation in the canonical form
we compute
where A and C are zero (why?)

34. Classification of second order PDEs: Example with variable coefficients
To obtain
And by solving (x, y) in terms of x and h we obtain:

35. Classification of second order PDEs: Summary of canonical forms
For a second order hyperbolic PDE in the form
by the change of parameter
We cast it into the 2nd canonical form:
Note: In fact for elliptic PDEs by the same form of transformation we can cast the PDE into elliptic canonical form:
For the derivation of this form and the parabolic canonical form refer to lesson 41 of [Farlow, 2012]:
Hyperbolic 1st
canonical form
Hyperbolic 2nd canonical form
Elliptic canonical form
Parabolic canonical form

36. Classification of second order PDEs: More than 2 independent variables
For a second order linear hyperbolic PDE with n independent variables:
Note a is expressed as symmetric matrix
Source: [Loret, 2008]

37. Classification of second order PDEs: More than 2 independent variables
Canonical form after coordinate transformation (refer to Loret chapter 3)
Elliptic:
Hyperbolic:
Parabolic:
Source: [Loret, 2008]

38. Classification of second order PDEs: More than 2 independent variables
Comparison with 2nd order PDE with two variables:
where
has the eigenvalues derived from

39. D’Alembert solution of the wave equation
Goals:
Obtain the solution to the wave equation
Solution of the PDE using characteristics from the PDE’s canonical form
The solution is,

40. D’Alembert solution of the wave equation Solution using the PDE’s canonical form
The characteristic parameters
cast the PDE into its canonical form
Now we can integrate the PDE on x and h respectively.
2 ODEs!

41. D’Alembert solution of the wave equation Use of initial conditions
By plugging in the initial conditions we want to solve the functions f and y:
By integrating the second equation we get
(K is set to zero because eventually in the solution of u K cancels out)

42. D’Alembert solution of the wave equation Final solution, left- and right- going waves
wave (speed c)
Left-going
wave (speed c)
Example: for the following IC:
The initial displacement is halved and propagated to the left and right with speeds c

43. Domain of influence and dependence Finite speed of information propagation
domain of dependence
Solution at x0, t0 only depends on the IC in [x0- c t0, x0+ c t0] which is called domain of dependence
Region where the solution of (x0, t0) influences is called domain of influence

44. Systems of 1st order PDEs (conservation laws) in 1D (2 independent parameters)

45. Systems of 1st order PDEs Characteristic values

46. Systems of 1st order PDEs Characteristic values

47. Systems of 1st order PDEs Eigenvalue problem for characteristic values

48. Systems of 1st order PDEs Initial conditions and solution for q

49. Systems of 1st order PDEs Solution process

50. Systems of 1st order PDEs Solution process

51. Systems of 1st order PDEs Example: 1D elastodynamics problem

52. Systems of 1st order PDEs Example: 1D elastodynamics problem

53. Systems of 1st order PDEs Example: 1D elastodynamics problem

54. Systems of 1st order PDEs 1D elastodynamics (no body force solution)

55. Systems of 1st order PDEs 1D elastodynamics (no body force solution)
domain of
influence

56. Systems of 1st order PDEs Hyperbolicity condition
Reminder: To solver the previous system of 1st order PDEs we should have been able to obtain matrix L for diagonalizing A in terms of L:
To obtain L (diagonalizing A) we should have
There are n linearly independent (left) eigenvectors
The corresponding eigenvalues li are real.
NOTE:
Hyperbolicity condition requires that we can find n characteristic values for the n-tuple q where information propagates along characteristics.

57. Geometric and algebraic multiplicity

58. Geometric and algebraic multiplicity

59. Geometric and algebraic multiplicity Symmetric tensors

60. Geometric and algebraic multiplicity Examples

61. Hyperbolicity of a system of 1st order PDEs Geometric and algebraic multiplicity

62. Systems of 1st order PDEs Hyperbolicity condition: Summary
(Also for semi-linear case)
Source: [LeVeque, 2002, 2.9]

63. Systems of 1st order PDEs Hyperbolicity condition: Summary
Quasi-linear system
where
Source: [LeVeque, 2002, 2.11]

64. Systems of 1st order PDEs More than 2 independent parameters (2D, 3D)

65. Systems of 1st order PDEs More than 2 independent parameters (2D, 3D)

66. Quasi-linear systems of 1st order PDEs Balance laws

67. Quasi-linear systems of 1st order PDEs Strong form of balance laws

68. Quasi-linear systems of 1st order PDEs Strong form of balance laws

69. Quasi-linear 1st order PDEs Formation of shocks & expansion wave
Motivation: Shock and expansion waves:
Example: Burger’s equation

70. Quasi-linear 1st order PDEs Formation of shocks & expansion wave
Example: Burger’s equation (continued)
Source: [Loret, 2008]

71. Quasi-linear 1st order PDEs Jump condition (brief overview)
Quasi-linear PDE (q(u))
We define the jump operator
where + and – refer to the two sides for the jump.
If Xs(t) is the position of the jump manifold in time, its equation is given by
This is called the jump or Rankine-Hugoniot condition.
Source: [Loret, 2008]

72. FYI Balance laws in spacetime (graphical view)
We can define spacetime flux by combining spatial flux fx with temporal flux ft (e.g. fx = -s, ft = p in elastodynamics)
The balance law is

73. FYI Jump condition (graphical view)
By writing two balance law expressions for W+ and W- we obtain the jump condition
nt is the spatial component of jump manifold, by 90˚ rotation from jump to normal direction we get the – sign.
We cannot define normal vectors in spacetime, but this sketch provides an idea how the jump condition is derived
Rankine-Hugoniot Jump conditions

74. Quasi-linear 1st order PDEs Jump formation example
Shock formation: example Traffic flow (u is speed)
Wave has speed of 1
Initial condition
Wave has speed of 0
Source: [Farlow, 2012]

75. Quasi-linear 1st order PDEs Jump formation example
For detailed solution derivation refer to [Farlow, 2012, lesson 28]
Solution:
1. Characteristics carrying u = 1 for for x0 ≤ 0.
2. Characteristics carrying 0 ≤ u ≤ 1 for 0 < x0 ≤ 1
3. Characteristics carrying u = 0 for 1 < x0
Source: [Farlow, 2012, lesson 28]

76. Quasi-linear 1st order PDEs Jump formation example
Shock formed as characteristics collide
S = (slope of shock)
Source: [Farlow, 2012, lesson 28]

77. Quasi-linear 1st order PDEs Jump formation example
Stages of solution
Source: [Farlow, 2012, lesson 28]

78. High temporal order of accuracy Motivation
Why important? Both temporal and spatial resolution affect the accuracy for wave propagation problems.
Typically methods are semi-discrete:
a form of integration such as Finite Difference is used for time integration with order q.
Very difficult and/or inefficient to achieve high temporal order of accuracy:
Runge-Kutta (RK) methods are rarely used for q > 4~6.
Achieving different temporal resolution for different elements is even more challenging!
tn+1 tn t x

79. Arbitrarily high temporal order of accuracy
Achieving high temporal orders in semi-discrete methods (CFEMs and DGs) is very challenging as the solution is only given at discrete times.
Perhaps the most successful method for achieving high order of accuracy in semi-discrete methods is the Taylor series of solution in time and subsequent use of Cauchy-Kovalewski or Lax-Wendroff procedure (FEM space derivatives  time derivatives). However, this method becomes increasingly challenging particularly for nonlinear problems.
High temporal order adversely affect stable time step size for explicit DG methods (e.g or worse for RKDG and ADER-DG methods).
Side note:
Spacetime (CFEM and DG) methods, on the other hand can achieve arbitrarily high temporal order of accuracy as the solution in time is directly discretized by FEM. (Discussed briefly in this course)

80. Asynchronous / no global time step
Geometry-induced stiffness results from simulating domains with drastically varying geometric features. Causes are:
Multiscale geometric features
Transition and boundary layers
Poor element quality (e.g. slivers)
Adaptive meshes driven by FEM discretization errors: e.g. crack tip singularities, moving wave fronts.
Time-marching methods
Time step is limited by smallest elements for explicit methods
Explicit: Efficient / stability concerns
Implicit: Unconditionally stable

81. Asynchronous / no global time step
Improvements:
Implicit-Explicit (IMEX) methods increase the time step by geometry splitting (implicit method for small elements) or operator splitting.
Local time-stepping (LTS): subcycling for smaller elements enables using larger global time steps
Example:
LTS by Dumbser, Munz, Toro, Lorcher, et. al.
Side note: SDGFEM (discussed later)
SDGFEM graciously and efficiently handles highly multiscale domains
Small elements locally have smaller progress in time (no global time step constrains)
None of the complicated “improvements” of time marching methods needed

82. Spacetime Discontinuous Galerkin (SDG) Finite Element Method
DG + spacetime meshing + causal meshes for hyperbolic problems:
Local solution property
O(N) complexity (solution cost scales linearly vs. number of elements N)
Asynchronous patch-by-patch solver
SDG
incoming characteristics on red boundaries
outgoing characteristics on green boundaries
The element can be solved as soon as inflow data on red boundary is obtained 
partial ordering & local solution property
elements of the same level can be solved in parallel
Time marching
Elements labeled 1 can be solved in parallel from initial conditions; elements 2 can be solved from their inflow element 1 solutions and so forth.
Time marching or the use of extruded meshes imposes a global coupling that is not intrinsic to a hyperbolic problem

83. Tent Pitcher: Causal spacetime meshing
Given a space mesh, Tent Pitcher constructs a spacetime mesh such that the slope of every facet on a sequence of advancing fronts is bounded by a causality constraint
Similar to CFL condition, except entirely local and not related to stability (required for scalability)
causality constraint
tent–pitching sequence
time

84. Tent Pitcher: Patch–by–patch meshing
meshing and solution are interleaved
patches (‘tents’) of tetrahedra are solve immediately  O(N) property
rich parallel structure: patches can be created and solved in parallel
tent–pitching sequence

85. A few other spacetime DG methods
Time Discontinuous Galerkin (TDG) methods (TJR Hughes, GM Hulbert 1987)
Elements are arranged in spacetime slabs
Discontinuities are only between spacetime slabs
Elements within slabs are solved simultaneously

86. A few other spacetime DG methods
Spacetime discontinuous Galerkin method (JJW Van der Vegt, H Van der Ven, et al)
Elements are arranged in spacetime slabs
Discontinuities are across all element boundaries
Elements within slabs are solved simultaneously as this method is implicit

87. Adaptive mesh operations
Local-effect adaptivity: no need for reanalysis of the entire domain
Arbitrary order and size in time:
Adaptive operations in spacetime:
ADER-DG with LTS
SDG
Example:
LTS by Dumbser, Munz, Toro, Lorcher, et. al.
Sod’s shock tube problem
Front-tracking better than shock capturing
hp-adaptivity better than h-adaptivity
Results by Scott Miller
Shock capturing: 473K elements
Shock tracking: 446 elements

88. 4. Adaptive mesh operations highly multiscale grids in spacetime
These meshes for a crack-tip wave scattering problem are generated by adaptive operations. Refinement ratio smaller than 10-4.
click to play movie
click to play movie
YouTube link
YouTube link
Color: log(strain energy); Height: velocity
Time in up direction

89. References
[Bathe, 2006] Bathe, K.-J. (2006). Finite element procedures. Klaus-Jurgen Bathe.
[Chapra and Canale, 2010] Chapra, S. C. and Canale, R. P. (2010). Numerical methods for engineers, volume 2. McGraw-Hill. 6th edition.
[Farlow, 2012] Farlow, S. J. (2012). Partial differential equations for scientists and engineers. Courier Corporation.
[Hughes, 2012] Hughes, T. J. (2012). The finite element method: linear static and dynamic finite element analysis. Courier Corporation.
[Levandosky, 2002] Levandosky, J. (2002). Math 220A, partial differential equations of applied mathematics, Stanford university. lecturenotes.html.
[LeVeque, 2002] LeVeque, R. L. (2002). Finite Volume Methods for Hyperbolic Problems. Cambridge University Press.
[Loret, 2008] Loret, B. (2008). Notes partial differential equations PDEs, institut national polytechnique de grenoble (inpg). loret_maths-EEE.html#TOP.
[Strikwerda, 2004] Strikwerda, J. C. (2004). Finite difference schemes and partial differential equations. SIAM.